If \(\rm \frac{{(1 - \cos \theta )}}{{\sin \theta }}\) = \(\frac{1}{5}\), then what will be the value of \(\rm \frac{{(1 + \cos \theta )}}{{\sin \theta }}\)?

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Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 02 Dec 2022 Shift 4)

Answer 1

Given:

\(\rm \frac{{(1 - \cos θ )}}{{\sin θ }}\) = \(\frac{1}{5}\)

Concept used:

cosec²θ - cot²θ = 1

cosec2θ - cot2θ = (cosecθ + cotθ)(cosecθ - cotθ)

Calculation:

\(\rm \frac{{(1 - \cos θ )}}{{\sin θ }}\) = \(\frac{1}{5}\)

⇒ \(\rm \frac{{1}}{{\sin θ }} - \frac{cos θ}{sinθ}\) = \(\frac{1}{5}\)

⇒ cosecθ - cotθ = \(\frac{1}{5}\)

Now,

(cosecθ + cotθ)(cosecθ - cotθ) = 1

⇒ (cosecθ + cotθ)\(\frac{1}{5}\) = 1

⇒ cosecθ + cotθ = 5

⇒ \(\rm \frac{{1}}{{\sin θ }} + \frac{cos θ}{sinθ}\) = 5

⇒ \(\rm \frac{{(1 + \cos \theta )}}{{\sin \theta }}\) = 5

∴ The required answer is 5.

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