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

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