Question

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

Answer 1

Answer with Step-by-step explanation:

Given : sinθ/(1− cosθ) = cosecθ + cotθ

L.H.S = sinθ/(1− cosθ)

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

[By rationalising ]

= sinθ(1+ cosθ)/(1− cos²θ)

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

=  [sinθ + (sinθ × cosθ)] /sin²θ

[By using the identity, (1- cos²θ) = sin²θ]

= ( sinθ /sin²θ ) + (sinθ × cosθ/sin²θ)

= 1/sinθ + cosθ/sinθ

=  cosec θ + cot θ

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

sinθ/(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..

Thanks