NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

The value of $\tan \left( \frac{7\pi}{8} \right)$ is

Choose the correct answer:

Explanation

1. Simplify the angle:

We can rewrite the angle $\frac{7\pi}{8}$ in terms of a known reference angle:

\tan \left( \frac{7\pi}{8} \right) = \tan \left( \pi - \frac{\pi}{8} \right)

Using the identity $\tan(\pi - \theta) = -\tan \theta$ :

\tan \left( \frac{7\pi}{8} \right) = -\tan \left( \frac{\pi}{8} \right)

2. Find the value of $\tan \left( \frac{\pi}{8} \right)$ :

Let $\theta = \frac{\pi}{8}$ , then $2\theta = \frac{\pi}{4}$ .

Using the half-angle formula $\tan \theta = \frac{1 - \cos 2\theta}{\sin 2\theta}$ :

\tan \left( \frac{\pi}{8} \right) = \frac{1 - \cos \frac{\pi}{4}}{\sin \frac{\pi}{4}}

\tan \left( \frac{\pi}{8} \right) = \frac{1 - \frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}

\tan \left( \frac{\pi}{8} \right) = \frac{\frac{\sqrt{2} - 1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}

\tan \left( \frac{\pi}{8} \right) = \sqrt{2} - 1

3. Final Calculation:

Substitute this back into our simplified expression:

\tan \left( \frac{7\pi}{8} \right) = -(\sqrt{2} - 1)

\tan \left( \frac{7\pi}{8} \right) = 1 - \sqrt{2}

Conclusion:

The value is $1 - \sqrt{2}$ . The correct option is (a).