Pythagorean identities are trig identities that come from the relationship \(\sin^2{x}+\cos^2{x}=1\). They are used to rewrite expressions involving sine, cosine, tangent, cotangent, secant, and cosecant. These problems focus on verifying identities by rewriting one side until it matches the other side.
Lesson
Notes
Questions
Verify the following.
\(\textbf{1)}\) \( \csc ^2 {x} (1-\cos ^2 {x}) = 1 \)
\(\textbf{2)}\) \( \tan ^2 {x} (\csc ^2 {x}-1) = 1 \)
\(\textbf{3)}\) \( \sec{x} – \cos{x} =\frac{\tan^2{x}}{\sec{x} } \)
\(\textbf{4)}\) \( \sin ^2 {x} -\cos ^2 {x} = 1-2\cos ^2 {x} \)
\(\textbf{5)}\) \( -\sec{x}\tan{x} = \frac{\csc{x}}{1-\csc^2{x}} \)
\(\textbf{6)}\) \( \tan ^2 {x} -\sin ^2 {x} =\tan ^2 {x} \sin ^2 {x} \)
\(\textbf{7)}\) \( \frac{\sin{x}\cos{x}}{(\sin{x}+\cos{x})^2-1}=\frac{1}{2} \)
\(\textbf{8)}\) \( \sin{x} + \csc{x}\cos^2{x} = \csc{x} \)
\(\textbf{9)}\) \( \cot ^2 {x} -\cos ^2 {x} =\cot ^2 {x} \cos ^2 {x} \)
\(\textbf{10)}\) \( (\sin{x}+\cos{x})^4 = (1+2\sin{x}\cos{x})^2 \)
\(\textbf{11)}\) \( \frac{\sec{x}}{\sec{x}-\cos{x}} = \csc^2{x} \)
\(\textbf{12)}\) \( (\csc{x}-\cot{x})(\cos{x}+1)=\sin{x} \)
\(\textbf{13)}\) \( 2\sin ^2 {x} -1=1-2\cos ^2 {x} \)
\(\textbf{14)}\) \( \sin ^4 {x} -\cos ^4 {x} =\sin ^2 {x} -\cos ^2 {x} \)
\(\textbf{15)}\) \( (1-\sin ^2 {x} )(1+\sin ^2 {x} )=1-\sin ^4 {x} \)
\(\textbf{16)}\) \( \frac{\tan{x}+\cot{x}}{\sin{x}\cos{x}} =\sec^2{x}+\csc^2{x} \)
\(\textbf{17)}\) \( \frac{1-\tan^2{x}}{1+\tan^2{x}}=\cos^2{x} -\sin^2{x} \)
\(\textbf{18)}\) \( \frac{1+\sec^2{x}}{1+\tan^2{x}}=1+\cos^2{x} \)
\(\textbf{19)}\) \( \tan ^2 {x} -\sin ^2 {x} =\tan ^2 {x} \sin ^2 {x} \)
\(\textbf{20)}\) \( \sec ^4 {x} -\tan ^4 {x} =\sec ^2 {x} +\tan ^2 {x} \)
See Related Pages\(\)
\(\bullet\text{ Right Triangle Trigonometry}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\)
\(\bullet\text{ Angle of Depression and Elevation}\)
\(\,\,\,\,\,\,\,\,\text{Angle of Depression}=\text{Angle of Elevation}…\)
\(\bullet\text{ Convert to Radians and to Degrees}\)
\(\,\,\,\,\,\,\,\,\text{Radians} \rightarrow \text{Degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\)
\(\bullet\text{ Degrees, Minutes and Seconds}\)
\(\,\,\,\,\,\,\,\,48^{\circ}34’21”…\)
\(\bullet\text{ Coterminal Angles}\)
\(\,\,\,\,\,\,\,\,\pm 360^{\circ} \text { or } \pm 2\pi n…\)
\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Find All 6 Trig Functions}\)
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\(…\)
\(\bullet\text{ Unit Circle}\)
\(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\)
\(\bullet\text{ Law of Sines}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\) 
\(…\)
\(\bullet\text{ Area of SAS Triangles}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{C}\) \(…\)
\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) \(…\)
\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) \(…\)
\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)
\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Inverse Trigonmetric Functions}\)
\(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\)
\(\bullet\text{ Sum and Difference of Angles Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}…\)
\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(2A)}=2\sin{(A)}\cos{(A)}…\)
\(\bullet\text{ Trigonometry-Pythagorean Identities}\)
\(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\)
\(\bullet\text{ Product-Sum Identities}\)
\(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\)
\(\bullet\text{ Cofunction Identities}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\)
\(\bullet\text{ Proving Trigonometric Identities}\)
\(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\)
\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
\(\,\,\,\,\,\,\,\,f(x)=A \sin{B(x-c)}+D \) 
\(…\)
\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)
\(\bullet\text{ Andymath Homepage}\)
