[Math] Express $\sec(2x)$ and $\tan(2x)$ in terms of $\tan x$

Question

$\tan(2x)$ was easy, as I just had to use the identity:

$$\tan(2x) = \frac{2\tan x}{1-\tan^2x}$$

I'm having a bit of trouble with $\sec(2x)$, however!

I tried squaring it and substituting a few other identities, but I either get stuck early on or go through cycles upon cycles of identity substitution which seems unnecessarily complicated.

Thanks in advance!

tan

Answer 1

Hint:

Start from $\;\cos 2x=2\cos^2x-1$ and use $1+\tan^2x=\dfrac1{\cos^2x}$.