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  4. The value of cos[ tan-1(tan(15π/4))] is

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Q. The value of $\cos\left[\tan^{-1}\left(tan\frac{15\pi}{4}\right)\right]$ is

Inverse Trigonometric Functions

Solution:

$tan^{-1} \left(tanx\right) = x$ if $-\frac{\pi}{2} < x <\frac{\pi}{2} $
$ tan \frac{15\pi}{4} = tan \left(4\pi-\frac{\pi}{4}\right) = tan \left(-\frac{\pi}{4}\right) $
$ \therefore tan^{-1} \left(tan \frac{15\pi}{4}\right) tan^{-1}\left(tan \left(-\frac{\pi}{4}\right)\right) = -\frac{\pi}{4}$
$ \therefore cos \left[tan^{-1}\left(tan \frac{15\pi}{4}\right) \right] = cos\left(-\frac{\pi}{4}\right) $
$ = cos \frac{\pi}{4} $
$ = \frac{1}{\sqrt{2}}$