Question

Prove the following trigonometric identities. $(tan\Theta+\frac{1}{cos\Theta})^{2}+(tan\Theta-\frac{1}{cos\Theta})^{2} =2(\frac{1+sin^{2}\Theta }{1-sin^{2}\Theta } )$

Answer 1

Answer WITH Step-by-step explanation:

Given : (tanθ + 1/cosθ)² + (tanθ − 1/cosθ)² = 2(1 + sin²θ/1 − sin²θ)

LHS :  (tanθ + 1/cosθ)² + (tanθ − 1/cosθ)²

= (tanθ + secθ)² + (tanθ − secθ)²

[By using, secθ = 1/ cosθ]

= tan²θ + sec²θ + 2tanθsecθ + tan²θ + sec²θ − 2tanθsecθ

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

= tan²θ + sec²θ + tan²θ + sec²θ + 2tanθsecθ − 2tanθsecθ

= 2tan²θ + 2sec²θ

= 2[tan²θ + sec²θ]

= 2[sin²θ/cos²θ + 1/cos²θ]

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

= 2[(1 + sin²θ)/cos²θ)]

[By taking LCM]

= 2[(1 + sin²θ)/(1 - sin²θ)]

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

(tanθ + 1/cosθ)² + (tanθ − 1/cosθ)² = 2(1 + sin²θ/1 − sin²θ)

L.H.S = R.H.S  

Hence Proved..

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Answer 2

Answer:

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