Find the best solution of the exercise 8.4 question no. 5(1) of introduction of trigonometry of ncert class 10 please help me to find out the easiest way to solve it .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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# 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). Q.5(1)

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(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)

^{2}The above equation is in the form of (a-b)

^{2}, 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.