Question

b. \cot x (\sec x + \tan x) = \csc x + 1

Explanation:

Step1: Expand left-hand side

$\cot x(\sec x + \tan x) = \cot x \sec x + \cot x \tan x$

Step2: Rewrite in sine/cosine

$\cot x = \frac{\cos x}{\sin x}$, $\sec x = \frac{1}{\cos x}$, $\tan x = \frac{\sin x}{\cos x}$
Substitute: $\frac{\cos x}{\sin x} \cdot \frac{1}{\cos x} + \frac{\cos x}{\sin x} \cdot \frac{\sin x}{\cos x}$

Step3: Simplify each term

First term: $\frac{\cos x}{\sin x \cos x} = \frac{1}{\sin x} = \csc x$
Second term: $\frac{\cos x \sin x}{\sin x \cos x} = 1$

Step4: Combine simplified terms

$\csc x + 1$

Answer:

The identity $\cot x(\sec x + \tan x) = \csc x + 1$ is verified.