What am I doing wrong when trying to find $\cos\tan^{-1}\frac{15}x$?

Question

I'm able to show that $\cos\tan^{-1}x= \frac1{x^2+1}$. But when doing the same with $x\to\frac{15}x$ I get the wrong answer.

Answer 1

Some algebraic manipulations ($x>0$):

$$\frac{1}{\sqrt{\frac{225}{x^2}+1}} = \frac{1}{\sqrt{\frac{1}{x^2}\left( 255+x^2\right)}} = \frac{1}{\frac{1}{x}\sqrt{ 255+x^2}} = \ldots$$