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  4. If m and M are the least and the greatest values of (cos-1x)2 + (sin-1+x)2, then (M/m) =

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Q. If m and $M$ are the least and the greatest values of $\left(cos^{-1}x\right)^{2} + \left(sin^{-1}+x\right)^{2}$, then $\frac{M}{m} = $

Inverse Trigonometric Functions

Solution:

$\left(\frac{\pi}{2}-sin^{-1}\,x\right)^{2}+ \left(sin^{-1}x\right)^{2}$
$= \frac{\pi^{2}}{8} + 2\left[sin^{-1}x-\frac{\pi}{4}\right]^{2}$
$m = \frac{\pi ^{2}}{8}$,
$M = \frac{5\pi ^{2}}{4}$,
$\frac{M}{m} = 10$