Prove the following identities:If $ \operatorname{cosec} \theta+\cot \theta=m $ and $ \operatorname{cosec} \theta-\cot \theta=n $, prove that $ m n=1 $

Given:

\( \operatorname{cosec} \theta+\cot \theta=m \) and \( \operatorname{cosec} \theta-\cot \theta=n \)

To do:

We have to prove that \( mn=1 \).

Solution:

We know that,

$\operatorname{cosec}^2 \theta-\cot^2 \theta=1$

$(a+b)(a-b)=a^{2}-b^{2}$

Therefore,

$m \times n = (\operatorname{cosec} \theta+\cot \theta)(\operatorname{cosec} \theta-\cot \theta)$

$=\operatorname{cosec}^{2} \theta-\cot ^{2} \theta$
$=1$

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Updated on: 10-Oct-2022

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