Exercises - Trigonometric Identities

  1. Show whether each of the following equations is or is not an identity:

    1.   $\displaystyle{\frac{\sin \theta}{\cos \theta} = 1 - \frac{\cos \theta}{\sin \theta}}$

    2.   $\displaystyle{1 - \cos^4 \theta = (2 - \sin^2 \theta) \sin^2 \theta}$

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

    4.   $\displaystyle{\frac{\sec \theta - \csc \theta}{\sec \theta + \csc \theta} = \frac{\tan \theta + 1}{\tan \theta - 1}}$

    5.   $\displaystyle{\frac{\sec^4 t - \tan^4 t}{1 - 2\tan^2 t} = 1}$

    6.   $\displaystyle{\sin^2 \theta \cot^2 \theta + \cos^2 \theta \tan^2 \theta = 1}$

    7.   $\displaystyle{\sec \theta - \frac{\cos \theta}{1 + \sin \theta} = \cot \theta}$

    8.   $\displaystyle{\frac{\tan^2 x}{1 + \cos x} = \frac{\sec x - 1}{\cos x}}$

    9.   $\displaystyle{(\csc t - \cot t)^2 = \frac{1 - \cos t}{1 + \cos t}}$

    10.   $\displaystyle{1 + \frac{1}{\cos \theta} = \frac{\tan^2 \theta}{\sec \theta - 1}}$

    11.   $\displaystyle{\frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1} = \frac{1 + \sin \theta}{\cos \theta}}$

    1. not an identity

    2. identity

    3. identity

    4. not an identity

    5. not an identity

    6. identity

    7. not an identity

    8. identity

    9. identity

    10. identity

    11. identity