Question

sintheta/1-cos theta=

Answer 1

Cosec theta+ cot theta

Answer 2

Answer:

$\frac{\sin\theta}{1-\cos\theta}=\csc\theta+\cot\theta$

Step-by-step explanation:

Given : $\frac{\sin\theta}{1-\cos\theta}$

We have to find an equivalent relation.

Consider the given expression $\frac{\sin\theta}{1-\cos\theta}$

Rationalize by multiply and divide by $1+\cos\theta$

The expression becomes,

$\frac{\sin\theta}{1-\cos\theta}\times \frac{1+\cos\theta}{1+\cos\theta}$

Simplify, we get,

$\frac{\sin\theta(1+\cos\theta)}{(1-\cos\theta)(1+\cos\theta)}$

Simplify denominator, using algebraic identity  $a^2-b^2=(a+b)(a-b)$

We have,

$\frac{\sin\theta(1+\cos\theta)}{(1-\cos^2\theta)}$

Using trigonometric identity $\sin^2\theta+\cos^2\theta=1$ , we have,

$\frac{\sin\theta(1+\cos\theta)}{\sin^2\theta}$

Simplify, we get,

$\frac{(1+\cos\theta)}{\sin\theta}$

We know

$\frac{1}{\sin\theta}=\csc\theta\\\\ \text{and}\\\ \frac{\cos\theta}{\sin\theta}=\cot\theta$

We get,

$\csc\theta+\cot\theta$

Thus, $\frac{\sin\theta}{1-\cos\theta}=\csc\theta+\cot\theta$