Question

Prove the following trigonometric identities:
1-cosA/1+cosA=(cota-cosecA)²

Answer 1

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Given❇

(1– cosA)/(1 + cosA) = (cot A – cosecA)²

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LHS : (1– cosA)/(1+ cosA)

= [(1− cosA)(1− cosA)]/[(1+ cosA)(1− cosA)]

[By Rationalising]

= (1− cosA)²/(1− cos²A)

= (1 + cos² A - 2cosA)/sin²A

[By using identity , (a + b)² = a² + 2ab + b² & (1- cos²θ) = sin²θ]

= 1/sin²A + cos² A/sin²A - 2cosA/sin²A

= cosec²A + cot²A - 2cosA/sinA × 1/sinA

[By using the identity, cosecθ = 1/sinθ, cotA = cosA/sinA ]

= cosec²A + cot²A - 2cotA × cosecA

= (cot A – cosecA)²

[By using identity , a² - 2ab + b² = (a - b)² ]

(1– cosA)/(1 + cosA) = (cot A – cosecA)²

L.H.S = R.H.S  

Hence Proved

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