The value of $cos \begin{Bmatrix} tan^{-1}\left(tan\frac{15\pi}{4}\right)\end{Bmatrix}$, is

$\frac{1}{\sqrt{2}}$

We have,

$cos \begin{Bmatrix} tan^{-1}\left(tan\frac{15\pi}{4}\right)\end{Bmatrix}$

$=cos \begin{Bmatrix} tan^{-1}\left(tan\left(4 \pi -\frac{\pi}{4}\right)\right)\end{Bmatrix}$

$=cos \begin{Bmatrix} tan\left(tan^{-1}\left( -\frac{\pi}{4}\right)\right)\end{Bmatrix}= cos \left(-\frac{\pi}{4}\right)= cos \frac{\pi}{4}=\frac{1}{\sqrt{2}}$