Q:

Find the value of $\cos\left({\tan^{-1}\left(\frac{3}{4}\right)}\right)$.

Find the value of $$\cos\left({\tan^{-1}\left(\dfrac{3}{4}\right)}\right)$$

A:

From $\sin^2(x)+ \cos^2(x)=1$ if you devide by $\cos^2$ and rearrange, you get
$$
\cos^2(x)=\frac{1}{\tan^2 (x)+1}
$$
then set $x=\tan^{-1}(y)$, taking the square root yelds
$$
\cos(\tan^{-1}(y))=\frac{1}{\sqrt{y^2+1}}
$$
In case $y=3/4$, so $\cos(\tan^{-1}(3/4))=4/5$

