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UPSC NDA II 2023 Mathematics Solved Paper

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Question : 26 of 120

Marks: +1, -0

What is 2 cot

(\frac{1}{2}cos^{-1}\frac{\sqrt{5}}{3})

equal to ?

$-1$
1
$3 + \sqrt{5}$
$3 - \sqrt{5}$

Solution:

Concept:

Use the half-angle identity for cotangent and find sine from cosine.

Explanation:

Let

\theta = \cos^{-1} \frac{\sqrt{5}}{3}

.

Then

\cos \theta = \frac{\sqrt{5}}{3}

.

Since

\theta

lies in

[0, \pi]

,

\sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \frac{5}{9}} = \sqrt{\frac{4}{9}} = \frac{2}{3}

.

Using the identity

\cot \frac{\theta}{2} = \frac{1 + \cos \theta}{\sin \theta}

, we get:

\cot \frac{\theta}{2} = \frac{1 + \frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{\frac{3 + \sqrt{5}}{3}}{\frac{2}{3}} = \frac{3 + \sqrt{5}}{3} \times \frac{3}{2} = \frac{3 + \sqrt{5}}{2}

.

Now,

2 \cot \frac{\theta}{2} = 2 \times \frac{3 + \sqrt{5}}{2} = 3 + \sqrt{5}

.

Answer:

3 + \sqrt{5}

(Option C).

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