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UPSC CDS 2 2022 Math Paper

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Question : 57 of 100

Marks: +1, -0

If

\frac{1}{cosec\theta -cot\theta }-\frac{1}{sin\theta }=x

\frac{1}{cosec\theta +cot\theta }-\frac{1}{sin\theta }

equal to, where 0 < θ <

\frac{\pi }{2}

?

-x
x
$\frac{1}{x}$
$-\frac{1}{x}$

Solution:

Concept:

Use the identity

\csc^2\theta - \cot^2\theta = 1

and rationalize denominators.

Explanation:

Given:

\frac{1}{\csc\theta - \cot\theta} - \frac{1}{\sin\theta} = x

.

Rationalize the first term:

\frac{1}{\csc\theta - \cot\theta} \cdot \frac{\csc\theta + \cot\theta}{\csc\theta + \cot\theta} = \frac{\csc\theta + \cot\theta}{\csc^2\theta - \cot^2\theta}

.

Since

\csc^2\theta - \cot^2\theta = 1

, this simplifies to

\csc\theta + \cot\theta

.

Also,

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

.

So,

(\csc\theta + \cot\theta) - \csc\theta = x

, hence

x = \cot\theta

.

Now evaluate

\frac{1}{\csc\theta + \cot\theta} - \frac{1}{\sin\theta}

.

Rationalize:

\frac{1}{\csc\theta + \cot\theta} \cdot \frac{\csc\theta - \cot\theta}{\csc\theta - \cot\theta} = \frac{\csc\theta - \cot\theta}{\csc^2\theta - \cot^2\theta} = \csc\theta - \cot\theta

.

Then

(\csc\theta - \cot\theta) - \csc\theta = -\cot\theta = -x

.

Answer:

-x

(Option A).

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