Question

Find y' if x = tan(y)

Answer 1

Use the chain rule:

$x=\tan y$
$\dfrac{\mathrm d}{\mathrm dx}x=\dfrac{\mathrm d}{\mathrm dx}\tan y$
$1=\sec^2y\dfrac{\mathrm dy}{\mathrm dx}$
$1=\sec^2y\,y'$
$y'=\dfrac1{\sec^2y}=\cos^2y$

Furthermore, if you restrict the domain of $y$, we could write

$x=\tan y\iff \tan^{-1}x=y$

and so we could also have

$y'=\cos^2(\tan^{-1}x)=\left(\dfrac1{\sqrt{x^2+1}}\right)^2=\dfrac1{x^2+1}$