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Section 5.2 – Verifying Trigonometric Identities

Section Details.
  • Proving trigonometric identities using known trigonometric identities and properties


Practice Problems


Directions. The following are review problems for the section. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. 
Verify the following identities.
  1. \(\displaystyle{\frac{3\cot^3t}{\csc t}=3\cos t(\csc^2t-1)}\) 

    \(\begin{align}
    \dfrac{3\cot^3t}{\csc t} &= \dfrac{3\cot t\cdot(\csc^2t-1)}{\csc t}\\
     &=\dfrac{3\dfrac{\cos t}{\sin t}\cdot(\csc^2t-1)}{\dfrac{1}{\sin t}}\\
      &=3\cos t\cdot(\csc^2t-1)\\
    \end{align}\)


    To see the full video page and find related videos, click the following link.
    WIR6 20B M150 V08


  2. \(\displaystyle{\tan x - \cot x = \sec x(2\sin x-\csc x)}\)

    \(\begin{align}
    \tan x - \cot x&=\dfrac{\sin x}{\cos x}-\dfrac{\cos x}{\sin x}\\
    &=\dfrac{\sin^2x-\cos^2x}{\sin x\cos x}\\
    &=\dfrac{\sin^2x-(1-\sin^2x)}{\sin x\cos x}\\
    &=\dfrac{2\sin^2x-1}{\sin x\cos x}\\
    &=\dfrac{2\sin^2x-1}{\sin x}\cdot\dfrac{1}{\cos x}\\
    &=\left(2\sin x-\dfrac{1}{\sin x}\right)\cdot\sec x\\
    &=\left(2\sin x-\csc x\right)\cdot\sec x\\
    \end{align}\)


    To see the full video page and find related videos, click the following link.
    WIR6 20B M150 V09