Exercises - Trigonometric Identities

  1. Determine if each of the below is an identity.

    1.     $\displaystyle{\sec^2 \theta + \csc^2 \theta = \sec^2 \theta \csc^2 \theta}$  

    2.     $\displaystyle{\sin t \cos t = \frac{1}{\tan t + \cot t}}$  

    3.     $\displaystyle{\frac{1+\tan t}{\tan t} = \cot t + \sec^2 t - \tan^2 t}$  

    4.     $\displaystyle{\frac{\tan^2 t - 1}{\sin t - \cos t} = \frac{\sin t - \cos t}{\cos^2 t}}$  

    5.     $\displaystyle{\frac{\cos 2\theta}{\sin \theta + \cos \theta} = \cos \theta - \sin \theta}$  

    6.     $\displaystyle{\cos 2\theta = 2\cos^2 \theta - 1}$  

    7.     $\displaystyle{1+ \sin t = \cos^3 t \tan t + \sin^3 t}$  

  2. Show whether each of the following is or is not an identity.

    1.     $\displaystyle{\frac{\sin \theta}{\cos \theta} = 1 - \frac{\cos \theta}{\sin \theta}}$   $\ans{\displaystyle{\textrm{this is not an identity}}}$

    2.     $\displaystyle{1 - \cos^4 \theta = (2 - \sin^2 \theta) \sin^2 \theta}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    3.     $\displaystyle{1 - 2\sin^2 \theta = 2 \cos^2 \theta - 1}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    4.     $\displaystyle{\frac{\sec \theta - \csc \theta}{\sec \theta + \csc \theta} = \frac{\tan \theta + 1}{\tan \theta - 1}}$   $\ans{\displaystyle{\textrm{this is not an identity}}}$

    5.     $\displaystyle{\frac{\sec^4 t - \tan^4 t}{1 - 2\tan^2 t} = 1}$   $\ans{\displaystyle{\textrm{this is not an identity}}}$

    6.     $\displaystyle{\sin^2 \theta \cot^2 \theta + \cos^2 \theta \tan^2 \theta = 1}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    7.     $\displaystyle{\sec \theta - \frac{\cos \theta}{1+\sin \theta} = \cot \theta}$   $\ans{\displaystyle{\textrm{this is not an identity}}}$

    8.     $\displaystyle{\frac{\tan^2 x}{1 + \cos x} = \frac{\sec x - 1}{\cos x}}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    9.     $\displaystyle{(\csc t - \cot t)^2 = \frac{1-\cos t}{1+ \cos t}}$   $\ans{\displaystyle{\textrm{this is an identity}}}$

    10.     $\displaystyle{1+ \frac{1}{\cos \theta} = \frac{ \tan^2 \theta}{\sec \theta - 1}}$   $\ans{\displaystyle{\textrm{this is an identity}}}$