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Problems

Verify the following.

\(\textbf{1)}\) \( \cos{⁡x} \sec{⁡x} = 1 \)

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\(\,\,\,\,\, \cos{⁡x} \sec{⁡x} = 1 \,\,\,\,\,\)\(\left(\text{Given}\right)\)

\(\,\,\,\,\, \cos{⁡x} \cdot \frac{1}{\cos{x}} = 1 \,\,\,\,\,\)\(\left(\sec{x}=\frac{1}{\cos{x}}\right)\)

\(\,\,\,\,\, 1 = 1 \,\,\,\,\,\)\(\left(\text{Inverse Property of Multiplication}\right)\)

\(\textbf{2)}\) \( \tan⁡{x} \cot{⁡x}=1 \)

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\(\,\,\,\,\, \tan{⁡x} \cot{⁡x} = 1 \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \frac{\sin{x}}{\cos{x}} \cdot \cot{x} = 1 \,\,\,\,\,\left(\tan{x}=\frac{\sin{x}}{\cos{x}}\right)\)

\(\,\,\,\,\, \frac{\sin{x}}{\cos{x}} \cdot \frac{\cos{x}}{\sin{x}} = 1 \,\,\,\,\,\left(\cot{x}=\frac{\cos{x}}{\sin{x}}\right)\)

\(\,\,\,\,\, 1 = 1 \,\,\,\,\,\left(\text{Inverse Property of Multiplication}\right)\)

\(\textbf{3)}\) \( \displaystyle \frac{\sin{x}}{\tan{x}} = \cos{x} \)

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\(\,\,\,\,\, \frac{\sin{x}}{\tan{x}} = \cos{x} \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \sin{x} \cdot \frac{1}{\tan{x}} = \cos{x} \,\,\,\,\,\left(\text{Split the fraction}\right)\)

\(\,\,\,\,\, \sin{x} \cdot \cot{x} = \cos{x} \,\,\,\,\,\left(\cot{x}=\frac{1}{\tan{x}}\right)\)

\(\,\,\,\,\, \sin{x} \cdot \frac{\cos{x}}{\sin{x}} = \cos{x} \,\,\,\,\,\left(\cot{x}=\frac{\cos{x}}{\sin{x}}\right)\)

\(\,\,\,\,\, \cos{x} = \cos{x} \,\,\,\,\,\left(\frac{\sin{x}}{\sin{x}}=1\right)\)

\(\textbf{4)}\) \( \displaystyle \sec{⁡x}+\tan{⁡x} = \frac{1+\sin{⁡x}}{\cos{⁡x}} \)

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\(\,\,\,\,\, \sec{⁡x}+\tan{⁡x} = \frac{1+\sin{⁡x}}{\cos{⁡x}} \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \frac{1}{\cos{x}}+\tan{⁡x} = \frac{1+\sin{⁡x}}{\cos{⁡x}} \,\,\,\,\,\left(\sec{x}=\frac{1}{\cos{x}}\right)\)

\(\,\,\,\,\, \frac{1}{\cos{x}}+\frac{\sin{x}}{\cos{x}} = \frac{1+\sin{⁡x}}{\cos{⁡x}} \,\,\,\,\,\left(\tan{x}=\frac{\sin{x}}{\cos{x}}\right)\)

\(\,\,\,\,\, \frac{1+\sin{⁡x}}{\cos{⁡x}} = \frac{1+\sin{⁡x}}{\cos{⁡x}} \,\,\,\,\,\left(\text{Combine Fractions with Common Denominator}\right)\)

\(\textbf{5)}\) \( \cos⁡{x} \csc⁡{x} + \tan{x} = \sec{x} \csc{⁡x} \)

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\(\,\,\,\,\, \cos⁡{x} \csc⁡{x} + \tan{x} = \sec{x} \csc{⁡x} \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \cos⁡{x} \cdot \frac{1}{\sin{x}} + \tan{x} = \sec{x} \csc{⁡x} \,\,\,\,\,\left(\csc{x}=\frac{1}{\sin{x}}\right)\)

\(\,\,\,\,\, \cos⁡{x} \cdot \frac{1}{\sin{x}} + \frac{\sin{x}}{\cos{x}} = \sec{x} \csc{⁡x} \,\,\,\,\,\left(\tan{x}=\frac{\sin{x}}{\cos{x}}\right)\)

\(\,\,\,\,\, \frac{\cos{x}}{\sin{x}} + \frac{\sin{x}}{\cos{x}} = \sec{x} \csc{⁡x} \,\,\,\,\,\left(\text{Multiply Fractions}\right)\)

\(\,\,\,\,\, \frac{\cos{x} \cdot \cos{x}}{\sin{x} \cdot \cos{x}} + \frac{\sin{x} \cdot \sin{x}}{\cos{x} \cdot \sin{x}} = \sec{x} \csc{⁡x} \,\,\,\,\,\left(\text{Common Denominators}\right)\)

\(\,\,\,\,\, \frac{\cos^2{x}}{\sin{x} \cos{x}} + \frac{\sin^2{x}}{\cos{x} \sin{x}} = \sec{x} \csc{⁡x} \,\,\,\,\,\left(\text{Simplify}\right)\)

\(\,\,\,\,\, \frac{\cos^2{x}+\sin^2{x}}{\sin{x} \cos{x}} = \sec{x} \csc{⁡x} \,\,\,\,\,\left(\text{Combine Fractions}\right)\)

\(\,\,\,\,\, \frac{1}{\sin{x} \cos{x}} = \sec{x} \csc{⁡x} \,\,\,\,\,\left(\sin^2{x}+\cos^2{x}=1\right)\)

\(\,\,\,\,\, \frac{1}{\cos{x}} \cdot \frac{1}{\sin{x}} = \sec{x} \csc{⁡x} \,\,\,\,\,\left(\text{Split into two fractions}\right)\)

\(\,\,\,\,\, \sec{x} \cdot \frac{1}{\sin{x}} = \sec{x} \csc{⁡x} \,\,\,\,\,\left(\frac{1}{\cos{x}}=\sec{x}\right)\)

\(\,\,\,\,\, \sec{x} \cdot \csc{x} = \sec{x} \csc{⁡x} \,\,\,\,\,\left(\frac{1}{\sin{x}}=\csc{x}\right)\)

\(\textbf{6)}\) \( \sec⁡{x}-\tan{⁡x}\sin⁡{x} = \cos{⁡x} \)

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\(\,\,\,\,\, \sec⁡{x}-\tan{⁡x}\sin⁡{x} = \cos{⁡x} \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \frac{1}{\cos{x}}-\tan{⁡x}\sin⁡{x} = \cos{⁡x} \,\,\,\,\,\left(\sec{x}=\frac{1}{\cos{x}}\right)\)

\(\,\,\,\,\, \frac{1}{\cos{x}}-\frac{\sin{x}}{\cos{x}}\sin⁡{x} = \cos{⁡x} \,\,\,\,\,\left(\tan{x}=\frac{\sin{x}}{\cos{x}}\right)\)

\(\,\,\,\,\, \frac{1}{\cos{x}}-\frac{\sin^2{x}}{\cos{x}} = \cos{⁡x} \,\,\,\,\,\left(\text{Multiply Fractions}\right)\)

\(\,\,\,\,\, \frac{1-\sin^2{x}}{\cos{x}} = \cos{⁡x} \,\,\,\,\,\left(\text{Combine Fractions with Common Denominator}\right)\)

\(\,\,\,\,\, \frac{\cos^2{x}}{\cos{x}} = \cos{⁡x} \,\,\,\,\,\left(\sin^2{x}+\cos^2{x}=1\right)\)

\(\,\,\,\,\, \cos{x} = \cos{⁡x} \,\,\,\,\,\left(\frac{\cos{x}}{\cos{x}}=1\right)\)

\(\textbf{7)}\) \( \tan{⁡x}+\cot⁡{x}=\sec⁡{x}\csc{⁡x} \)

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\(\,\,\,\,\, \tan{⁡x}+\cot⁡{x}=\sec⁡{x}\csc{⁡x} \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \frac{\sin{x}}{\cos{x}}+\cot⁡{x}=\sec⁡{x}\csc{⁡x} \,\,\,\,\,\left(\tan{x}=\frac{\sin{x}}{\cos{x}}\right)\)

\(\,\,\,\,\, \frac{\sin{x}}{\cos{x}}+\frac{\cos{x}}{\sin{x}}=\sec⁡{x}\csc{⁡x} \,\,\,\,\,\left(\cot{x}=\frac{\cos{x}}{\sin{x}}\right)\)

\(\,\,\,\,\, \frac{\sin{x}}{\sin{x}}\cdot\frac{\sin{x}}{\cos{x}}+\frac{\cos{x}}{\sin{x}}\cdot\frac{\cos{x}}{\cos{x}}=\sec⁡{x}\csc{⁡x} \,\,\,\,\,\left(\text{Common Denominators}\right)\)

\(\,\,\,\,\, \frac{\sin^2{x}}{\cos{x}\sin{x}}+\frac{\cos^2{x}}{\cos{x}\sin{x}}=\sec⁡{x}\csc{⁡x} \,\,\,\,\,\left(\text{Simplify}\right)\)

\(\,\,\,\,\, \frac{\sin^2{x}+\cos^2{x}}{\cos{x}\sin{x}}=\sec⁡{x}\csc{⁡x} \,\,\,\,\,\left(\text{Combine Fractions with Common Denominator}\right)\)

\(\,\,\,\,\, \frac{1}{\cos{x}\sin{x}}=\sec⁡{x}\csc{⁡x} \,\,\,\,\,\left(\text{Pythagorean Identities } \sin^2{x}+\cos^2{x}=1 \right)\)

\(\,\,\,\,\, \frac{1}{\cos{x}}\cdot\frac{1}{\sin{x}}=\sec⁡{x}\csc{⁡x} \,\,\,\,\,\left(\text{Split into two fractions} \right)\)

\(\,\,\,\,\, \sec{x}\cdot\frac{1}{\sin{x}}=\sec⁡{x}\csc{⁡x} \,\,\,\,\,\left(\sec{x}=\frac{1}{\cos{x}}\right)\)

\(\,\,\,\,\, \sec{x}\cdot\csc{x}=\sec⁡{x}\csc{⁡x} \,\,\,\,\,\left(\csc{x}=\frac{1}{\sin{x}}\right)\)

\(\textbf{8)}\) \( \csc ^2 {x} (1-\cos ^2 {x}) = 1 \)

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\(\,\,\,\,\, \csc ^2 {x} (1-\cos ^2 {x}) = 1 \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \frac{1}{\sin^2{x}} (1-\cos ^2 {x}) = 1 \,\,\,\,\,\left(\csc^2{x}=\frac{1}{\sin^2{x}}\right)\)

\(\,\,\,\,\, \frac{1}{\sin^2{x}} (\sin ^2 {x}) = 1 \,\,\,\,\,\left(1-\cos^2{x}=\sin^2{x}\right)\)

\(\,\,\,\,\, 1 = 1 \,\,\,\,\,\left(\frac{\sin^2{x}}{\sin^2{x}}=1\right)\)

\(\textbf{9)}\) \( \tan ^2 {x} (\csc ^2 {x}-1) = 1 \)

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\(\,\,\,\,\, \tan ^2 {x} (\csc ^2 {x}-1) = 1 \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \tan ^2 {x} \cdot \cot ^2 {x} = 1 \,\,\,\,\,\left(\csc^2{x}-1=cot^2{x}\right)\)

\(\,\,\,\,\, \tan ^2 {x} \cdot \displaystyle \frac{1}{tan^2{x}} = 1 \,\,\,\,\,\left(\cot^2{x}=\frac{1}{tan^2{x}}\right)\)

\(\,\,\,\,\, 1 = 1 \,\,\,\,\,\left(\frac{\tan^2{x}}{\tan^2{x}}=1\right)\)

\(\textbf{10)}\) \( \displaystyle \frac{\sin{x}}{\tan{x}} + \frac{\cos{x}}{\cot{x}} = \sin{x} + \cos{x} \)

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\(\,\,\,\,\, \frac{\sin{x}}{\tan{x}} + \frac{\cos{x}}{\cot{x}} = \sin{x} + \cos{x} \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \sin{x} \cdot \frac{1}{\tan{x}} + \cos{x} \cdot \frac{1}{\cot{x}} = \sin{x} + \cos{x} \,\,\,\,\,\left(\text{Split the fractions}\right)\)

\(\,\,\,\,\, \sin{x} \cdot \cot{x} + \cos{x} \cdot \frac{1}{\cot{x}} = \sin{x} + \cos{x} \,\,\,\,\,\left(\cot{x}=\frac{1}{\tan{x}}\right)\)

\(\,\,\,\,\, \sin{x} \cdot \cot{x} + \cos{x} \cdot \tan{x} = \sin{x} + \cos{x} \,\,\,\,\,\left(\tan{x}=\frac{1}{\cot{x}}\right)\)

\(\,\,\,\,\, \sin{x} \cdot \frac{\cos{x}}{\sin{x}} + \cos{x} \cdot \tan{x} = \sin{x} + \cos{x} \,\,\,\,\,\left(\cot{x}=\frac{\cos{x}}{\sin{x}}\right)\)

\(\,\,\,\,\, \sin{x} \cdot \frac{\cos{x}}{\sin{x}} + \cos{x} \cdot \frac{\sin{x}}{cos{x}} = \sin{x} + \cos{x} \,\,\,\,\,\left(\tan{x}=\frac{\sin{x}}{\cos{x}}\right)\)

\(\,\,\,\,\, \cos{x} + \sin{x} = \sin{x} + \cos{x} \,\,\,\,\,\left(\text{Simplify}\right)\)

\(\,\,\,\,\, \sin{x} + \cos{x} = \sin{x} + \cos{x} \,\,\,\,\,\left(\text{Commutative Property}\right)\)

\(\textbf{11)}\) \( \displaystyle \sec{x} – \cos{x} =\frac{\tan^2{x}}{\sec{x} } \)

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\(\,\,\,\,\, \displaystyle \sec{x} – \cos{x} =\frac{\tan^2{x}}{\sec{x} } \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \displaystyle \sec{x} – \cos{x} =\frac{\sec^2{x}-1}{\sec{x} } \,\,\,\,\,\left(\tan^2{x}=\sec^2{x}-1\right)\)

\(\,\,\,\,\, \displaystyle \sec{x} – \cos{x} =\frac{\sec^2{x}}{\sec{x} }-\frac{1}{\sec{x} } \,\,\,\,\,\left(\text{Split the fraction}\right)\)

\(\,\,\,\,\, \displaystyle \sec{x} – \cos{x} =\sec{x}-\frac{1}{\sec{x} } \,\,\,\,\,\left(\frac{sec^2{x}}{\sec{x}}=\sec{x}\right)\)

\(\,\,\,\,\, \displaystyle \sec{x} – \cos{x} =\sec{x}-\cos{x}\,\,\,\,\,\left(\cos{x}=\frac{1}{\sec{x}}\right)\)

\(\textbf{12)}\) \( \sin ^2 {x} -\cos ^2 {x} = 1-2\cos ^2 {x} \)

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\(\,\,\,\,\, \sin ^2 {x} -\cos ^2 {x} = 1-2\cos ^2 {x} \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, 1- \cos ^2 {x} -\cos ^2 {x} = 1-2\cos ^2 {x} \,\,\,\,\,\left(\sin ^2 {x}=1-\cos ^2 {x}\right)\)

\(\,\,\,\,\, 1- 2\cos ^2 {x} = 1-2\cos ^2 {x} \,\,\,\,\,\left(\cos ^2 {x}+\cos ^2 {x} =2\cos ^2 {x}\right)\)

\(\textbf{13)}\) \( \displaystyle -\sec{x}\tan{x} = \frac{\csc{x}}{1-\csc^2{x}} \)

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\(\,\,\,\,\, \displaystyle -\sec{x}\tan{x} = \frac{\csc{x}}{1-\csc^2{x}} \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \displaystyle -\sec{x}\tan{x} = \frac{\csc{x}}{-\cot^2{x}} \,\,\,\,\,\left(1-\csc^2{x}=-\cot^2{x}\right)\)

\(\,\,\,\,\, \displaystyle -\sec{x}\tan{x} = -\frac{\csc{x}}{1} \cdot \frac{1}{\cot^2{x}} \,\,\,\,\,\left(\text{Split the fractions}\right)\)

\(\,\,\,\,\, \displaystyle -\sec{x}\tan{x} = -\frac{1}{\sin{x}} \cdot \frac{1}{\cot^2{x}} \,\,\,\,\,\left(\csc{x}=\frac{1}{\sin{x}}\right)\)

\(\,\,\,\,\, \displaystyle -\sec{x}\tan{x} = -\frac{1}{\sin{x}} \cdot tan^2{x} \,\,\,\,\,\left(\tan^2{x}=\frac{1}{\cot^2{x}}\right)\)

\(\,\,\,\,\, \displaystyle -\sec{x}\tan{x} = -\frac{1}{\sin{x}} \cdot \frac{\sin^2{x}}{\cos^2{x}} \,\,\,\,\,\left(\tan^2{x}=\frac{\sin^2{x}}{\cos^2{x}}\right)\)

\(\,\,\,\,\, \displaystyle -\sec{x}\tan{x} = – \frac{\sin{x}}{\cos^2{x}} \,\,\,\,\,\left(\frac{\sin{x}}{\cos{x}}=1\right)\)

\(\,\,\,\,\, \displaystyle -\sec{x}\tan{x} = – \frac{1}{\cos{x}} \cdot \frac{\sin{x}}{\cos{x}} \,\,\,\,\,\left(\text{Split the fraction}\right)\)

\(\,\,\,\,\, \displaystyle -\sec{x}\tan{x} = – \sec{x} \cdot \frac{\sin{x}}{\cos{x}} \,\,\,\,\,\left(\frac{1}{\cos{x}}=\sec{x}\right)\)

\(\,\,\,\,\, \displaystyle -\sec{x}\tan{x} = – \sec{x} \cdot \tan{x} \,\,\,\,\,\left(\frac{\sin{x}}{\cos{x}}=\tan{x}\right)\)

\(\textbf{14)}\) \( \tan ^2 {x} -\sin ^2 {x} =\tan ^2 {x} \sin ^2 {x} \)

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\(\,\,\,\,\, \tan ^2 {x} -\sin ^2 {x} =\tan ^2 {x} \sin ^2 {x} \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \displaystyle \frac{\sin^2{x}}{\cos^2{x}} -\sin ^2 {x} =\tan ^2 {x} \sin ^2 {x} \,\,\,\,\,\left(\tan^2{x}=\frac{\sin^2{x}}{\cos^2{x}}\right)\)

\(\,\,\,\,\, \displaystyle \sin ^2 {x} \left(\frac{1}{\cos^2{x}} -1 \right) =\tan ^2 {x} \sin ^2 {x} \,\,\,\,\,\left(\text{Factor out } \sin^2{x}\right)\)

\(\,\,\,\,\, \displaystyle \sin ^2 {x} \left( \sec^2{x} -1 \right) =\tan ^2 {x} \sin ^2 {x} \,\,\,\,\,\left(\sec^2{x}=\frac{1}{\cos^2{x}}\right)\)

\(\,\,\,\,\, \displaystyle \tan^2{x} \sin ^2 {x} =\tan ^2 {x} \sin ^2 {x} \,\,\,\,\,\left(\text{Commutative Property of Multiplication}\right)\)

\(\textbf{15)}\) \( \displaystyle \frac{\sin{x}\cos{x}}{(\sin{x}+\cos{x})^2-1}=\frac{1}{2} \)

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\(\,\,\,\,\, \displaystyle \frac{\sin{x}\cos{x}}{(\sin{x}+\cos{x})^2-1}=\frac{1}{2} \,\,\,\,\,\left(\text{Given}\right)\)


\(\,\,\,\,\, \displaystyle \frac{\sin{x}\cos{x}}{(\sin{x}+\cos{x})(\sin{x}+\cos{x})-1}=\frac{1}{2} \,\,\,\,\,\left(\left(a+b\right)^2=\left(a+b\right)\left(a+b\right)\right)\)


\(\,\,\,\,\, \displaystyle \frac{\sin{x}\cos{x}}{\sin^2{x}+2\sin{x}\cos{x}+\cos^2{x}-1}=\frac{1}{2} \,\,\,\,\,\left(\left(a+b\right)\left(a+b\right)=a^2+2ab+b^2\right)\)


\(\,\,\,\,\, \displaystyle \frac{\sin{x}\cos{x}}{2\sin{x}\cos{x}+1-1}=\frac{1}{2} \,\,\,\,\,\left(\sin^2{x}+\cos^2{x}=1\right)\)


\(\,\,\,\,\, \displaystyle \frac{\sin{x}\cos{x}}{2\sin{x}\cos{x}}=\frac{1}{2} \,\,\,\,\,\left(1-1=0\right)\)


\(\,\,\,\,\, \displaystyle \frac{1}{2}=\frac{1}{2} \,\,\,\,\,\left(\frac{\sin{x}\cos{x}}{\sin{x}\cos{x}}=1\right)\)

\(\textbf{16)}\) \( \sin{x} + \csc{x}\cos^2{x} = \csc{x} \)

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\(\,\,\,\,\, \sin{x} + \csc{x}\cos^2{x} = \csc{x} \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \sin{x} + \frac{1}{\sin{x}} \cdot \cos^2{x} = \csc{x} \,\,\,\,\,\left(\csc{x}=\frac{1}{\sin{x}}\right)\)

\(\,\,\,\,\, \frac{\sin{x}}{\sin{x}} \cdot \frac{\sin{x}}{1} + \frac{1}{\sin{x}} \cdot \cos^2{x} = \csc{x} \,\,\,\,\,\left(\frac{\sin{x}}{\sin{x}}=1\right)\)

\(\,\,\,\,\, \frac{\sin^2{x}}{\sin{x}} + \frac{\cos^2{x}}{\sin{x}} = \csc{x} \,\,\,\,\,\left(\sin{x} \cdot \sin{x}= \sin^2{x}\right)\)

\(\,\,\,\,\, \frac{\sin^2{x}+\cos^2{x}}{\sin{x}} = \csc{x} \,\,\,\,\,\left(\text{Fractions with common denominator}\right)\)

\(\,\,\,\,\, \frac{1}{\sin{x}} = \csc{x} \,\,\,\,\,\left(\sin^2{x}+\cos^2{x}=1\right)\)

\(\,\,\,\,\, \csc{x} = \csc{x} \,\,\,\,\,\left(\frac{1}{\sin{x}} = \csc{x}\right)\)

\(\textbf{17)}\) \( \cot ^2 {x} -\cos ^2 {x} =\cot ^2 {x} \cos ^2 {x} \)

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\(\,\,\,\,\, \cot ^2 {x} -\cos ^2 {x} =\cot ^2 {x} \cos ^2 {x} \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \frac{\cos^2{x}}{\sin^2{x}} -\cos ^2 {x} =\cot ^2 {x} \cos ^2 {x} \,\,\,\,\,\left(\cot^2{x}=\frac{\cos^2{x}}{\sin^2{x}}\right)\)

\(\,\,\,\,\, \cos^2{x}\left(\frac{1}{\sin^2{x}} -1\right) =\cot ^2 {x} \cos ^2 {x} \,\,\,\,\,\left(\text{Factor out } \cos^2{x}\right)\)

\(\,\,\,\,\, \cos^2{x}\left(\csc^2{x} -1\right) =\cot ^2 {x} \cos ^2 {x} \,\,\,\,\,\left(\csc^2{x}=\frac{1}{\sin^2{x}}\right)\)

\(\,\,\,\,\, \cos^2{x}\left(\cot^2{x}\right) =\cot ^2 {x} \cos ^2 {x} \,\,\,\,\,\left(\cot^2{x}=\csc^2{x}-1\right)\)

\(\,\,\,\,\, \cot^2{x}\cos^2{x} =\cot ^2 {x} \cos ^2 {x} \,\,\,\,\,\left(\text{Commutative Property of Multiplication}\right)\)

\(\textbf{18)}\) \( (\sin{x}+\cos{x})^4 = (1+2\sin{x}\cos{x})^2 \)

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\(\,\,\,\,\, (\sin{x}+\cos{x})^4 = (1+2\sin{x}\cos{x})^2 \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \left((\sin{x}+\cos{x})^2\right)^2 = (1+2\sin{x}\cos{x})^2 \,\,\,\,\,\left(x^4=\left(x^2\right)^2\right)\)

\(\,\,\,\,\, \left((\sin{x}+\cos{x})(\sin{x}+\cos{x})\right)^2 = (1+2\sin{x}\cos{x})^2 \,\,\,\,\,\left(\left(a+b\right)^2=\left(a+b\right)\left(a+b\right)\right)\)

\(\,\,\,\,\, \left(\sin^2{x}+2\sin{x}\cos{x}+\cos^2{x}\right)^2 = (1+2\sin{x}\cos{x})^2 \,\,\,\,\,\left(\left(a+b\right)\left(a+b\right)=a^2+2ab+b^2\right)\)

\(\,\,\,\,\, \left(1+2\sin{x}\cos{x}\right)^2 = \left(1+2\sin{x}\cos{x}\right)^2 \,\,\,\,\,\left(\sin^2{x}+2\sin{x}\cos{x}+\cos^2{x}\right)\)

\(\textbf{19)}\) \( \displaystyle \frac{\sec{x}}{\sec{x}-\cos{x}} = \csc^2{x} \)

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\(\,\,\,\,\, \displaystyle \frac{\sec{x}}{\sec{x}-\cos{x}} = \csc^2{x} \,\,\,\,\,\left(\text{Given}\right)\)

\(\,\,\,\,\, \displaystyle \frac{\sec{x}}{\frac{1}{\cos{x}}}-\frac{\cos{x}}{1} = \csc^2{x} \,\,\,\,\,\left(\sec{x}=\frac{1}{\cos{x}}\right)\)

\(\,\,\,\,\, \displaystyle \frac{\sec{x}}{\frac{1}{\cos{x}}}-\frac{\cos{x}}{1} \cdot \frac{\cos{x}}{\cos{x}} = \csc^2{x} \,\,\,\,\,\left(\frac{\cos{x}}{\cos{x}}=1\right)\)

\(\,\,\,\,\, \displaystyle \frac{\sec{x}}{\frac{1}{\cos{x}}}- \frac{\cos^2{x}}{\cos{x}} = \csc^2{x} \,\,\,\,\,\left(\cos{x}\cos{x}=\cos^2{x}\right)\)

\(\,\,\,\,\, \displaystyle \frac{\sec{x}}{\frac{1-\cos^2{x}}{\cos{x}}} = \csc^2{x} \,\,\,\,\,\left(\text{Fractions with common denominators}\right)\)

\(\,\,\,\,\, \displaystyle \frac{\sec{x}}{\frac{sin^2{x}}{\cos{x}}} = \csc^2{x} \,\,\,\,\,\left(\sin^2{x}=1-\cos^2{x}\right)\)

\(\,\,\,\,\, \displaystyle \frac{1}{\cos{x} \cdot \frac{sin^2{x}}{\cos{x}}} = \csc^2{x} \,\,\,\,\,\left(\sec{x}=\frac{1}{\cos{x}}\right)\)

\(\,\,\,\,\, \displaystyle \frac{1}{sin^2{x}} = \csc^2{x} \,\,\,\,\,\left(\frac{\cos{x}}{\cos{x}}=1\right)\)

\(\,\,\,\,\, \displaystyle \csc^2{x} = \csc^2{x} \,\,\,\,\,\left(\csc{x}=\frac{1}{\sin{x}}=1\right)\)

\(\textbf{20)}\) \( (\csc{x}-\cot{x})(\cos{x}+1)=\sin{x} \)

\(\textbf{21)}\) \( 2\sin ^2 {x} -1=1-2\cos ^2 {x} \)

\(\textbf{22)}\) \( \sin ^4 {x} -\cos ^4 {x} =\sin ^2 {x} -\cos ^2 {x} \)

\(\textbf{23)}\) \( (1-\sin ^2 {x} )(1+\sin ^2 {x} )=1-\sin ^4 {x} \)

\(\textbf{24)}\) \( \displaystyle \frac{\tan{x}+\cot{x}}{\sin{x}\cos{x}} =\sec^2{x}+\csc^2{x} \)

\(\textbf{25)}\) \( \displaystyle \frac{1-\tan^2{x}}{1+\tan^2{x}}=\cos^2{x} -\sin^2{x} \)

\(\textbf{26)}\) \( \displaystyle \frac{1+\sec^2{x}}{1+\tan^2{x}}=1+\cos^2{x} \)

\(\textbf{27)}\) \( \displaystyle \frac{\sin{x} +\cos{x} }{\sec{x} +\csc{x} }= \sin{x} \cos{x} \)

\(\textbf{28)}\) \( \displaystyle \frac{\csc{x} +\sec{x} }{\cot{x} +\tan{x} }= \sin{x} +\cos{x} \)

\(\textbf{29)}\) \( \displaystyle \frac{1-\cos{x}}{\sin{x} }+\frac{\sin{x}}{1-\cos{x}}= 2\csc{x} \)

\(\textbf{30)}\) \( \displaystyle \frac{\cot{x} -\csc{x} }{1-\sec{x} }=\cot{x} \)

\(\textbf{31)}\) \( \tan ^2 {x} -\sin ^2 {x} =\tan ^2 {x} \sin ^2 {x} \)

\(\textbf{32)}\) \( \sec ^4 {x} -\tan ^4 {x} =\sec ^2 {x} +\tan ^2 {x} \)

\(\textbf{33)}\) \( \cos ^2 {x} -cos ^4 {x} =\cos ^2 {x} \sin ^2 {x} \)

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\(\,\,\,\,\,\, \cos ^2 {x} -cos ^4 {x} =\cos ^2 {x} \sin ^2 {x} \,\,\,\,\, \left(\text{Given}\right) \)

\(\,\,\,\,\,\, \cos ^2 {x}\left( 1 -cos ^2 {x}\right) =\cos ^2 {x} \sin ^2 {x} \,\,\,\,\, \left(\text{Factor out } \cos^2 {x}\right) \)

\(\,\,\,\,\,\, \cos ^2 {x}\left( \sin^2 {x}\right) =\cos ^2 {x} \sin ^2 {x} \,\,\,\,\, \left(1-\cos^2{x}=\sin^2 {x}\right) \)

\(\,\,\,\,\,\, \cos ^2 {x} \sin^2 {x} =\cos ^2 {x} \sin ^2 {x} \,\,\,\,\, \left(\text{Simplify}\right) \)

\(\textbf{34)} \) \( \displaystyle \frac{\sin{2x}}{\sin{x}}-\frac{\cos{2x}}{\cos{x}}=\sec{x}\)

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\(\,\,\,\,\,\displaystyle\frac{\sin{2x}}{\sin{x}}-\frac{\cos{2x}}{\cos{x}}=\sec{x}\,\,\,\,\, \left(\text{Given}\right) \)


\(\,\,\,\,\,\displaystyle\frac{2\sin{x}\cos{x}}{\sin{x}}-\frac{2\cos^2{x}-1}{\cos{x}}=\sec{x}\,\,\,\,\, \left(\text{Double Angle Formulas}\right) \)


\(\,\,\,\,\,\displaystyle\frac{2\cos{x}}{1}-\frac{2\cos^2{x}-1}{\cos{x}}=\sec{x}\,\,\,\,\, \left(\text{Simplify}\right) \)


\(\,\,\,\,\,\displaystyle\frac{2\cos^2{x}}{\cos{x}}-\frac{2\cos^2{x}-1}{\cos{x}}=\sec{x}\,\,\,\,\, \left(\text{Common Denominator}\right) \)


\(\,\,\,\,\,\displaystyle\frac{2\cos^2{x}-\left(2\cos^2{x}-1\right)}{\cos{x}}=\sec{x}\,\,\,\,\, \left(\text{Combine into 1 fraction}\right) \)


\(\,\,\,\,\,\displaystyle\frac{2\cos^2{x}-2\cos^2{x}+1}{\cos{x}}=\sec{x}\,\,\,\,\, \left(\text{Distribute the negative}\right) \)


\(\,\,\,\,\,\displaystyle\frac{1}{\cos{x}}=\sec{x}\,\,\,\,\, \left(\text{Simplify}\right) \)


\(\,\,\,\,\,\sec{x}=\sec{x}\,\,\,\,\, \left(\text{Reciprocal Identities}\right) \)

\(\textbf{35)} \) \( \displaystyle \frac{1-\tan^2{x}}{1+\tan^2{x}}=\cos{2x}\)

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\(\,\,\,\,\,\displaystyle\frac{1-\tan^2{x}}{1+\tan^2{x}}=\cos{2x}\,\,\,\,\, \left(\text{Given}\right) \)


\(\,\,\,\,\,\displaystyle\frac{1-\tan^2{x}}{\sec^2{x}}=\cos{2x}\,\,\,\,\, \left(\text{Pythagorean Identities}\right) \)


\(\,\,\,\,\,\displaystyle\frac{\cos^2{x}\left(1-\tan^2{x}\right)}{1}=\cos{2x}\,\,\,\,\, \left(\text{Reciprocal Identities}\right) \)


\(\,\,\,\,\,\cos^2{x}-\cos^2{x}\tan^2{x}=\cos{2x}\,\,\,\,\, \left(\text{Distribution}\right) \)


\(\,\,\,\,\,\cos^2{x}-\cos^2{x}\frac{\sin^2{x}}{\cos^2{x}}=\cos{2x}\,\,\,\,\, \left(\text{Quotient Identities}\right) \)


\(\,\,\,\,\,\cos^2{x}-\sin^2{x}=\cos{2x}\,\,\,\,\, \left(\text{Simplify}\right)\)


\(\,\,\,\,\,\cos{2x}=\cos{2x}\,\,\,\,\, \left(\text{Double Angle Formulas}\right)\)

\(\textbf{36)}\) \(\displaystyle \frac{\sin ^2{x}}{1-\cos{x}}= \cos{x}+1\)

\(\textbf{37)}\) \(\displaystyle \left(\cot{x}\right)\left(\cot{x}+\tan{x}\right)= \csc ^2{x}\)

\(\textbf{38)}\) \(\displaystyle \frac{1+\tan ^2{x}}{\csc ^2{x}}= \tan ^2{x}\)

\(\textbf{39)}\) \(\displaystyle \cos ^2{x}+\tan ^2{x}\cos ^2{x}=1\)

See Related Pages\(\)

\(\bullet\text{ Right Triangle Trigonometry}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\)

\(\bullet\text{ Angle of Depression and Elevation}\)
\(\,\,\,\,\,\,\,\,\text{Angle of Depression}=\text{Angle of Elevation}…\)

\(\bullet\text{ Convert to Radians and to Degrees}\)
\(\,\,\,\,\,\,\,\,\text{Radians} \rightarrow \text{Degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\)

\(\bullet\text{ Degrees, Minutes and Seconds}\)
\(\,\,\,\,\,\,\,\,48^{\circ}34’21”…\)

\(\bullet\text{ Coterminal Angles}\)
\(\,\,\,\,\,\,\,\,\pm 360^{\circ} \text { or } \pm 2\pi n…\)

\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Find All 6 Trig Functions}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Unit Circle}\)
\(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\)

\(\bullet\text{ Law of Sines}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\) \(…\)

\(\bullet\text{ Area of SAS Triangles}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{C}\) \(…\)

\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) \(…\)

\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) \(…\)

\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)

\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Inverse Trigonmetric Functions}\)
\(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\)

\(\bullet\text{ Sum and Difference of Angles Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}…\)

\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(2A)}=2\sin{(A)}\cos{(A)}…\)

\(\bullet\text{ Trigonometry-Pythagorean Identities}\)
\(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\)

\(\bullet\text{ Product-Sum Identities}\)
\(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\)

\(\bullet\text{ Cofunction Identities}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\)

\(\bullet\text{ Proving Trigonometric Identities}\)
\(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\)

\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
\(\,\,\,\,\,\,\,\,f(x)=A \sin{B(x-c)}+D \) \(…\)

\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)

In Summary

Trigonometry identities show all the unique ways that trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant interact with each other.

One The most commonly used trigonometry identities are the Pythagorean Identities.

\(\sin^2 \theta + \cos^2 \theta = 1\)
\(1 + \tan^2 \theta = \sec^2 \theta\)
\(1+ \cot^2 \theta = \csc^2 \theta\)

Here is a video explaining how these are derived.

There are many other useful trigonometric identities. Try the examples above to see how they can be used to verify other trigonometric identities.

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