Some Applications of Trigonometry

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Question

Prove the following identities where the angles involved are acute angles for which the expressions are defined.
$space space left parenthesis cosecθ space minus space cotθ right parenthesis squared space equals space fraction numerator 1 minus cosθ over denominator 1 plus cosθ end fraction.$

Solution

left parenthesis cosecθ minus cotθ right parenthesis squared space equals space fraction numerator 1 minus cosθ over denominator 1 plus cosθ end fraction

L.H.S. =

left parenthesis cosecθ space minus space cotθ right parenthesis squared

open parentheses 1 over sinθ minus cosθ over sinθ close parentheses squared

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equals space fraction numerator left parenthesis 1 minus cosθ right parenthesis squared over denominator sin squared straight theta end fraction equals space fraction numerator left parenthesis 1 minus cosθ right parenthesis squared over denominator 1 minus cos squared straight theta end fraction

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Hence, L.H.S. = R.H.S.

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Some Applications Of Trigonometry

Prove the following identities where the angles involved are acute angles for which the expressions are defined.
$space space left parenthesis cosecθ space minus space cotθ right parenthesis squared space equals space fraction numerator 1 minus cosθ over denominator 1 plus cosθ end fraction.$

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