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UPSC NDA Math Model Paper 10

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Question : 55 of 120

Marks: +1, -0

What is the value of

\frac{\sin\theta + \cos\theta - \tan\theta}{\sec\theta + \csc\theta - \cot\theta}

\theta = \frac{3\pi}{4} ?

0
1
-1
None of the above

Solution:

Here,

\frac{\sin\theta + \cos\theta - \tan\theta}{\sec\theta + \csc\theta - \cot\theta}

and

\theta = \frac{3\pi}{4} = \pi - \frac{\pi}{4}

\frac{\sin\theta + \cos\theta - \tan\theta}{\sec\theta + \csc\theta - \cot\theta}

=\frac{\sin\left(\pi - \frac{\pi}{4}\right) + \cos\left(\pi - \frac{\pi}{4}\right) - \tan\left(\pi - \frac{\pi}{4}\right)}{\sec\left(\pi - \frac{\pi}{4}\right) + \csc\left(\pi - \frac{\pi}{4}\right) - \cot\left(\pi - \frac{\pi}{4}\right)}

=\frac{\sin\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{4}\right) + \tan\left(\frac{\pi}{4}\right)}{-\sec\left(\frac{\pi}{4}\right) + \csc\left(\frac{\pi}{4}\right) + \cot\left(\frac{\pi}{4}\right)}

= \frac{\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} + 1}{-\sqrt{2} + \sqrt{2} + 1}

= 1

Hence, option (2) is correct.

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