Question

Prove the following trignometric identities. $1+\frac{cot^{2}\Theta }{1+cosec\Theta} =cosec\Theta$

Answer 1

Answer with Step-by-step explanation:

Given : 1 +  [cot²θ/ (1 + cosecθ)] = cosecθ

LHS : 1 +  [cot²θ/ (1 + cosecθ)]

= 1 + [(cosec²θ − 1) / (1 + cosecθ)]

[By using  an identity, cot²θ = cosec²θ −1]

= 1 + [(cosecθ − 1)(cosecθ + 1)] / (1 + cosecθ)

[By using identity , a² - b² =  (a + b) (a - b)]

= 1 + cosecθ − 1

= cosecθ

1 +  [cot²θ/ (1 + cosecθ)] = cosecθ

L.H.S = R.H.S  

Hence Proved..

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Answer 2

We know that, $cosec^{2}\theta - cot^{2}\theta = 1$

⇒ $\cot^{2}\theta = cosec^{2}\theta - 1 = (cosec \theta + 1)(cosec \theta - 1)$

⇒ $\frac{cot^{2}\theta}{1 + cosec \theta} = cosec \theta - 1$

⇒$1 + \frac{cot^{2}\theta}{1 + cosec \theta} = cosec \theta$

QED