Skill: Apply reciprocal and quotient identities

Reciprocal identities rewrite a trigonometric function as $1$ divided by another function:

$$\csc\theta=\frac{1}{\sin\theta},\qquad \sec\theta=\frac{1}{\cos\theta},\qquad \cot\theta=\frac{1}{\tan\theta}.$$

Quotient identities rewrite tangent and cotangent using sine and cosine:

$$\tan\theta=\frac{\sin\theta}{\cos\theta},\qquad \cot\theta=\frac{\cos\theta}{\sin\theta}.$$

These equations are used only when their denominators are nonzero. For example, $\tan\theta=\frac{\sin\theta}{\cos\theta}$ requires $\cos\theta\ne0$. Reciprocal functions such as $\sec\theta$ are not the same as inverse trigonometric functions such as $\arccos\theta$.

Worked example

Simplify

$$\frac{\tan\theta}{\sec\theta},$$

assuming $\cos\theta\ne0$.

Step 1: Apply the quotient identity to tangent.

$$\tan\theta=\frac{\sin\theta}{\cos\theta}$$

Step 2: Apply the reciprocal identity to secant.

$$\sec\theta=\frac{1}{\cos\theta}$$

Substitute both identities:

$$\frac{\tan\theta}{\sec\theta} = \frac{\frac{\sin\theta}{\cos\theta}}{\frac{1}{\cos\theta}}.$$

Step 3: Divide by a fraction by multiplying by its reciprocal.

$$\frac{\frac{\sin\theta}{\cos\theta}}{\frac{1}{\cos\theta}} = \frac{\sin\theta}{\cos\theta}\cdot\frac{\cos\theta}{1}.$$

Step 4: Cancel $\cos\theta$. This is valid because $\cos\theta\ne0$.

$$\frac{\sin\theta}{\cos\theta}\cdot\cos\theta = \sin\theta.$$

Therefore,

$$\boxed{\frac{\tan\theta}{\sec\theta}=\sin\theta}$$

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