Question

Prove the following trignometric identities. $\frac{cos\Theta}{cosec\Theta+1}+\frac{cos\Theta}{cosec\Theta-1} =2tan\Theta$

Answer 1

Answer with Step-by-step explanation:

Given :  cosθ / (cosecθ + 1) + cosθ / (cosecθ − 1) = 2tanθ

L.H.S = cosθ / (1/ sinθ + 1) + cosθ / ( 1/sinθ − 1)

[By using, cosecθ = 1/ sinθ]

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

[By taking LCM ]

= (cosθ) × (sinθ)/(1 + sinθ) + (cosθ) × (sinθ)/ (1 −sinθ)

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

[By taking LCM ]

= sinθcosθ − sin²θcosθ + sinθcosθ + sin² θcosθ / (1 - sin²θ)

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

= (2sinθcosθ) / cos²θ

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

= 2sinθ/cosθ

= 2tanθ

[By using the identity, tanθ = sinθ/cosθ ]

cosθ / (cosecθ + 1) + cosθ / (cosecθ − 1) = 2tanθ

L.H.S = R.H.S  

Hence Proved..

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Answer 2

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