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UPSC CDS II 2025 Mathematics Paper

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Question : 31 of 100

Marks: +1, -0

\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+....}}}}=cosec\theta

, then what is sin θ equal to?

1
$\frac{\sqrt{3}}{2}$
$\frac{1}{\sqrt{2}}$
$\frac{1}{2}$

Solution:

Concept:

An infinite nested radical of the form

\sqrt{k + \sqrt{k + \sqrt{k + \dots}}}

satisfies

x^2 = k + x

, where

x

is its value.

Explanation:

Let

x = \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}}

.

Then

x^2 = 2 + x

. Rearranging gives

x^2 - x - 2 = 0

.

Factorising:

(x - 2)(x + 1) = 0

. So

x = 2

or

x = -1

.

Since the radical is positive,

x = 2

.

Given

\sqrt{2 + \sqrt{2 + \dots}} = \csc \theta

, so

\csc \theta = 2

.

Using

\sin \theta = \frac{1}{\csc \theta}

, we get

\sin \theta = \frac{1}{2}

.

Answer:

\sin \theta = \frac{1}{2}

, which corresponds to option D.

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