Question

Prove the following trigonometric identities. $\frac{1+sin\Theta}{cos\Theta}+\frac{cos\Theta}{1+sin\Theta}=2sec\Theta$

Answer 1

Answer with Step-by-step explanation:

Given :  

(1+ sinθ)/ cosθ + cosθ /(1 + sinθ) = 2secθ

LHS = (1+ sinθ)/ cosθ + cosθ /(1 + sinθ)

= (1 + sinθ)² + cos²θ) / [cosθ(1+sinθ)]

[By taking LCM]

= (1 + sin²θ + 2sinθ + cos²θ)/[cosθ(1+sinθ)]

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

= ( sin²θ +cos²θ + 2sinθ + 1)/[cosθ(1+sinθ)]

= ( 1 + 2sinθ + 1)/[cosθ(1+sinθ)]

[By using the identity , sin² θ + cos² θ = 1]

= ( 2 + 2sinθ )/[cosθ(1+sinθ)]

= 2(1 + sinθ)/ [cosθ(1+sinθ)]

= 2/cosθ

= 2 × 1/cosθ

= 2 secθ

[By using the identity, secθ = 1/ cosθ]

L.H.S = R.H.S  

Hence Proved..

HOPE THIS ANSWER WILL HELP YOU...

Answer 2

Answer:

Refer to the attachment for your answer.