Skill: Verify fundamental trigonometric identities

To verify a trigonometric identity, transform one side only until it becomes the other side. Use known identities, algebraic operations, and common denominators. Also check where the expressions are defined.

Useful fundamental identities include

$$\sin^2\theta+\cos^2\theta=1$$ $$\sec\theta=\frac{1}{\cos\theta} \qquad\text{and}\qquad \tan\theta=\frac{\sin\theta}{\cos\theta}.$$

Example

Verify that

$$\sec\theta-\cos\theta=\frac{\sin^2\theta}{\cos\theta}.$$

First, note the denominator restriction:

$$\cos\theta\ne 0.$$

This is required because both $\sec\theta$ and $\dfrac{\sin^2\theta}{\cos\theta}$ are undefined when $\cos\theta=0$.

Start with the left-hand side:

$$\sec\theta-\cos\theta$$

Use the reciprocal identity $\sec\theta=\dfrac{1}{\cos\theta}$:

$$\frac{1}{\cos\theta}-\cos\theta$$

Rewrite $\cos\theta$ with denominator $\cos\theta$:

$$\frac{1}{\cos\theta}-\frac{\cos^2\theta}{\cos\theta}$$

Combine the fractions:

$$\frac{1-\cos^2\theta}{\cos\theta}$$

Using $\sin^2\theta+\cos^2\theta=1$, rewrite $1-\cos^2\theta$ as $\sin^2\theta$:

$$\frac{\sin^2\theta}{\cos\theta}$$

This matches the right-hand side, so

$$\boxed{\sec\theta-\cos\theta=\frac{\sin^2\theta}{\cos\theta}}$$

for all angles where $\cos\theta\ne0$.

An identity is true for every angle in its domain; it is not something that needs to be solved for particular angle values.