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Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (i) (cosec θ - cot θ)^2 = (1-cos θ)/(1+cos θ)

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3 months ago Anand Kumar Pandey
2803 Points
```							Dear Student(cosec θ-cot θ)^2= (1-cos θ)/(1+cos θ)To prove this, first take the Left-Hand side (L.H.S) of the given equation, to prove the Right Hand Side (R.H.S)L.H.S. = (cosec θ-cot θ)^2The above equation is in the form of (a-b) , and expand itSince (a-b)^2= a^2+ b^2–2abHere a = cosec θ and b = cot θ= (cosec^2 θ +cot^2 θ-2cosec θ cot θ)Now,apply the corresponding inverse functions and equivalent ratios to simplify= (1/sin^2 θ+ cos^2 θ/sin^2 θ-2cos θ/sin^2 θ)= (1 + cos^2 θ-2cos θ)/(1 - cos^2 θ)= (1-cos θ)^2 /(1 -cosθ)(1+cos θ)= (1-cos θ)/(1+cos θ) =R.H.S.Therefore,(cosec θ-cot θ)^2= (1-cos θ)/(1+cos θ)Hence proved.Thanks
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3 months ago
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