Question

Proving Identities

Verify the identity$\frac{\cos u\sec u}{\tan u}=\cot u$

Short answer

The expression$\frac{\cos u\sec u}{\tan u}=\cot u$ is an identity.

Step-by-step solution

Step 1. Given information.

An expression$\frac{\cos u\sec u}{\tan u}=\cot u$

Step 2. Concept used.

First divide the question into two sections LHS and RHS. After that simplify both separately, if the outcomes of both are same then it is an identity.

Step 3. Calculation.

Now, simplify LHS of $\frac{\cos u\sec u}{\tan u}=\cot u$:

We know that,

$\begin{array}{l}\tan u=\frac{\sin u}{\cos u}\\ \sec u=\frac{1}{\cos u}\end{array}$

$$\begin{gathered} \frac{\cos u\times \sec u}{\tan u}=\frac{\cos u\times \frac{1}{\cos u}}{\frac{\sin u}{\cos u}} \\ \frac{\cos u\times \sec u}{\tan u}=\frac{\cos u}{\sin u} \\ \frac{\cos u\times \sec u}{\tan u}=\cot u \end{gathered}$$

RHS of the equation is $\cot u$

Both RHS and LHS are equal. Hence it is an identity.