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Mathematics Question on Inverse Trigonometric Functions

The value of tan⁡−1[1+x2+1−x21+x2−1−x2],∣x∣<12,x≠0,\tan^{-1} \left[\frac{\sqrt{1+x^{2}} + \sqrt{1-x^{2}}}{\sqrt{1+x^{2}} - \sqrt{1-x^{2}}}\right] , \left|x\right| < \frac{1}{2}, x \ne0, is equal to :

A

π4+12cos⁡−1x2\frac{\pi}{4} + \frac{1}{2} \cos^{-1} x^{2}

B

π4+cos⁡−1x2\frac{\pi}{4} + \cos^{-1} x^{2}

C

π4−12cos⁡−1x2\frac{\pi}{4} - \frac{1}{2} \cos^{-1} x^{2}

D

π4−cos⁡−1x2\frac{\pi}{4} - \cos^{-1} x^{2}

Answer

π4+12cos⁡−1x2\frac{\pi}{4} + \frac{1}{2} \cos^{-1} x^{2}

Explanation

Solution

x2=cos2θx^{2} = cos 2\theta
θ=12cos−1x2\theta = \frac{1}{2} cos^{-1} x^{2}