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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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