### Question 11

SSC-CGL 2020 Mar 4 Shift 2

The value of $\displaystyle \frac{\tan^2\theta - \sin^2 \theta}{2 + \tan^2 \theta + \cot^2 \theta}$ is?

- $\displaystyle \sin^6 \theta$
- $\displaystyle \cos^4 \theta$
- $\displaystyle \tan^3 \theta$
- $\displaystyle \sec^5 \theta$

### Solution in Short

Choose an angle so that each of the given options gives a different value on that! Once such value is θ = 45°.

$\displaystyle \frac{\tan^2\theta - \sin^2 \theta}{2 + \tan^2 \theta + \cot^2 \theta}$

$\displaystyle = \frac{\tan^2 45 \degree - \sin^2 45 \degree}{2 + \tan^2 45 \degree + \cot^2 45 \degree}$

$\displaystyle = \frac{1 - 1/2}{2 + 1 + 1} = \frac 18$

We can verify that option (a) gives $\displaystyle \sin^6 45 \degree = 1/8$, and hence the correct answer!

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