Chapter 7.1, Problem 69E

To determine

To verify the identity:

  $\frac{\csc x-\cot x}{\sec x-1}=\cot x$

Explanation of Solution

Given:

An identity is given as

  $\frac{\csc x-\cot x}{\sec x-1}=\cot x$

Concept Used:

The trigonometric identities:

  $\begin{array}{l}\sec x=\frac{1}{\cos x}\\ \csc x=\frac{1}{\sin x}\\ \cot x=\frac{\cos x}{\sin x}\end{array}$

Calculation:

In order to verify the identity:

  $\frac{\csc x-\cot x}{\sec x-1}=\cot x$

Take left hand of above identity and simplify it using the trigonometric identities.

As,

  $\begin{array}{c}\frac{\csc x-\cot x}{\sec x-1}=\frac{\frac{1}{\sin x}-\frac{\cos x}{\sin x}}{\frac{1}{\cos x}-1}\\ =\frac{\frac{1-\cos x}{\sin x}}{\frac{1-\cos x}{\cos x}}\\ =\frac{1-\cos x}{\sin x}\times \frac{\cos x}{1-\cos x}\\ =\frac{\cos x}{\sin x}\\ =\cot x\\ =\text{Right hand}\end{array}$

Thus, identity is verified, i.e.

  $\frac{\csc x-\cot x}{\sec x-1}=\cot x$