Question

Prove the following trigonometric identities. $\sqrt{\frac{1-cos\Theta}{1+cos\Theta} }=cosec\Theta-cot\Theta$

Answer 1

Answer with Step-by-step explanation:

Given :  

√(1− cosθ)/(1 + cosθ) = cosecθ − cotθ

L.H.S : √(1− cosθ)/(1 + cosθ)

Multiplying both numerator and denominator by (1− cosθ),

= √(1− cosθ)/(1+cosθ)

= √[(1− cosθ)(1−cosθ)] / [(1+ cosθ)(1− cosθ)]

= √[(1−cosθ)²/(1−cos²θ)]

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

=√(1− cosθ)²/sin²θ

[By using the identity , 1−cos²θ = sin² θ ]

= (1− cosθ)/sinθ

= 1/sinθ− cosθ/sinθ

= cosecθ − cotθ

[By using the identity ,1 /sinθ = cosec θ & cosθ/sinθ = cotθ]

(1− cosθ)/(1 + cosθ) = cosecθ − cotθ

L.H.S = R.H.S  

Hence Proved..

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Answer 2

$\huge\bf{Hey\:Mate}$

Refer to the attatchment.

Hope it helps❤️