Question

1 = COSФ / 1- COSФ = ( COSECФ = COTФ)²
CORRECT AND CLEAR ANSWER WILL BE MARKED AS BRAINLIST

Answer 1

Given

$\bullet\ \; \rm \dfrac{1+cos\theta}{1-cos\theta}=(csc\theta+cot\theta)^2$

To Prove

$\bullet\ \; \rm \dfrac{1+cos\theta}{1-cos\theta}=(csc\theta+cot\theta)^2$

Solution

Take RHS

$\to \bf \pink{(csc\theta +cot\theta)^2}\\\\\to \rm \green{csc^2\theta+cot^2\theta+2.csc\theta.cot\theta}\\\\\to \rm \green{\dfrac{1}{sin^2\theta}+\dfrac{cos^2\theta}{sin^2\theta}+\dfrac{2}{sin\theta}.\dfrac{cos\theta}{sin\theta}}\\\\\to \rm \green{\dfrac{1}{sin^2\theta}+\dfrac{cos^2\theta}{sin^2\theta}+\dfrac{2cos\theta}{sin^2\theta}}\\\\\to \rm \green{\dfrac{1+cos^2\theta +2cos\theta}{sin^2\theta}}\\\\\to \rm \green{\dfrac{(1+cos\theta )^2}{1-cos^2\theta}}$

∵ ( a + b )² = a² + b² = 2ab

∵ sin²θ = 1 - cos²θ

$\to \rm \green{\dfrac{(1+cos\theta)^2}{(1+cos\theta)(1-cos\theta)}}$

∵ ( a + b ) ( a - b ) = a² - b²

$\to \bf \green{\dfrac{1+cos\theta}{1-cos\theta}}\\\\\bf RHS$

Hence Proved