Question

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Answer 1

Question:-

Prove that:

$\sf \large \frac{1 + sin \theta}{cos \theta} + \frac{cos \theta}{1 + sin \theta} = 2sec \theta.$

Given:-

$\sf \large Identity \: \: \sf \large \frac{1 + sin \theta}{cos \theta} + \frac{cos \theta}{1 + sin \theta} = 2sec \theta.$

To Prove:-

$\sf \large \frac{1 + sin \theta}{cos \theta} + \frac{cos \theta}{1 + sin \theta} = 2sec \theta.$

Solution:-

L.H.S.

LCM of (cosθ) and (1 + sinθ) is cosθ (1 + sinθ).

$\sf \large = \frac{(1 + sin \theta) {}^{2} + cos {}^{2} \theta}{(cos \theta)(1 + sin \theta)} .$

$\sf \large = \frac{1 + sin {}^{2} \theta + 2sin \theta + cos {}^{2} \theta }{cos \theta( 1 + sin \theta)} .$

$\sf \large = \frac{1 + 1 + 2sin \theta}{cos \theta( 1 + si n \theta)} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \large ( \because sin {}^{2} \theta + cos {}^{2} \theta = 1).$

$\sf \large = \frac{2(1 + sin \theta)}{cos \theta(1 + sin \theta)} = 2sec \theta \: \: \: \: R.H.S.$

Answer:-

${ \boxed{ \sf \large \red{ \sf \large \frac{1 + sin \theta}{cos \theta} + \frac{cos \theta}{1 + sin \theta} = 2sec \theta.}}}$

Hence, Proved

L.H.S. = R.H.S.

Hope you have satisfied. ⚘

Answer 2

Step-by-step explanation:

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