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Lesson

Notes

Questions

Verify the following.

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

Show Steps

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

\(\,\,\,\,\,\,\csc ^2 {x} (\sin ^2 {x}) = 1\)

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

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

\(\,\,\,\,\,\,1 = 1\)

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

\(\textbf{3)}\) \( \sec{x} – \cos{x} =\frac{\tan^2{x}}{\sec{x} } \)

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

\(\textbf{5)}\) \( -\sec{x}\tan{x} = \frac{\csc{x}}{1-\csc^2{x}} \)

\(\textbf{6)}\) \( \tan ^2 {x} -\sin ^2 {x} =\tan ^2 {x} \sin ^2 {x} \)

\(\textbf{7)}\) \( \frac{\sin{x}\cos{x}}{(\sin{x}+\cos{x})^2-1}=\frac{1}{2} \)

\(\textbf{8)}\) \( \sin{x} + \csc{x}\cos^2{x} = \csc{x} \)

\(\textbf{9)}\) \( \cot ^2 {x} -\cos ^2 {x} =\cot ^2 {x} \cos ^2 {x} \)

\(\textbf{10)}\) \( (\sin{x}+\cos{x})^4 = (1+2\sin{x}\cos{x})^2 \)

\(\textbf{11)}\) \( \frac{\sec{x}}{\sec{x}-\cos{x}} = \csc^2{x} \)

\(\textbf{12)}\) \( (\csc{x}-\cot{x})(\cos{x}+1)=\sin{x} \)

\(\textbf{13)}\) \( 2\sin ^2 {x} -1=1-2\cos ^2 {x} \)

\(\textbf{14)}\) \( \sin ^4 {x} -\cos ^4 {x} =\sin ^2 {x} -\cos ^2 {x} \)

\(\textbf{15)}\) \( (1-\sin ^2 {x} )(1+\sin ^2 {x} )=1-\sin ^4 {x} \)

\(\textbf{16)}\) \( \frac{\tan{x}+\cot{x}}{\sin{x}\cos{x}} =\sec^2{x}+\csc^2{x} \)

\(\textbf{17)}\) \( \frac{1-\tan^2{x}}{1+\tan^2{x}}=\cos^2{x} -\sin^2{x} \)

\(\textbf{18)}\) \( \frac{1+\sec^2{x}}{1+\tan^2{x}}=1+\cos^2{x} \)

\(\textbf{19)}\) \( \tan ^2 {x} -\sin ^2 {x} =\tan ^2 {x} \sin ^2 {x} \)

\(\textbf{20)}\) \( \sec ^4 {x} -\tan ^4 {x} =\sec ^2 {x} +\tan ^2 {x} \)

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\(\bullet\text{ Andymath Homepage}\)

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