Question

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

Answer 1

Answer with Step-by-step explanation:

Given : cosθ/1−sinθ = 1+ sinθ / cosθ

L.H.S : cosθ/1−sinθ

Multiplying both numerator and denominator by  (1 + sinθ), we have

= cosθ(1+ sinθ)/[(1− sinθ)(1 + sinθ)]

= cosθ (1+ sinθ)/(1− sin²θ)

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

=  cosθ(1 + sinθ)/cos²θ

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

= (1+ sinθ)/cosθ

cosθ/1−sinθ = (1 + sinθ) / cosθ

L.H.S = R.H.S  

Hence Proved..

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Answer 2

SOLUTION

L.H.S

$= > \frac{cos \theta}{1 - sin \theta} \\ = > \frac{cos \theta}{1 - sin \theta} \times \frac{1 + sin \theta}{1 + sin \theta} \\ \\ = > \frac{cos \theta(1 + sin \theta)}{1 - {sin}^{2} \theta} \\ \\ = > \frac{cos \theta(1 + sin \theta)}{ {cos}^{2} \theta} \\ \\ = > \frac{1 + sin \theta}{cos \theta}$

R.H.S.

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