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UPSC CDS 2 2022 Math Paper

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Question : 54 of 100

Marks: +1, -0

If x =

\frac{1+sin\theta }{cos\theta }

\frac{tan\theta +sec\theta -1}{tan\theta -sec\theta +1}

equal to ?

-x
x
2x
$\frac{x}{2}$

Solution:

Concept:

Simplify the given expression using the identity

1 = \sec^2\theta - \tan^2\theta

.

Explanation:

Let

E = \frac{\tan\theta + \sec\theta - 1}{\tan\theta - \sec\theta + 1}

.

Substitute

1 = \sec^2\theta - \tan^2\theta

in the numerator:

E = \frac{\tan\theta + \sec\theta - (\sec^2\theta - \tan^2\theta)}{\tan\theta - \sec\theta + 1}

.

Factor the difference of squares:

\sec^2\theta - \tan^2\theta = (\sec\theta - \tan\theta)(\sec\theta + \tan\theta)

.

So numerator becomes

(\sec\theta + \tan\theta) - (\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = (\sec\theta + \tan\theta)[1 - (\sec\theta - \tan\theta)]

.

Thus

E = \frac{(\sec\theta + \tan\theta)[1 - \sec\theta + \tan\theta]}{\tan\theta - \sec\theta + 1}

.

Notice that

1 - \sec\theta + \tan\theta = \tan\theta - \sec\theta + 1

(same as denominator).

Cancel the common factor:

E = \sec\theta + \tan\theta

.

Now,

\sec\theta + \tan\theta = \frac{1}{\cos\theta} + \frac{\sin\theta}{\cos\theta} = \frac{1+\sin\theta}{\cos\theta} = x

.

Answer:

x

(Option B).

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