Recall the definitions:
- sec(x)=1/\cos(x)
- \tan(x)=\sin(x)/\cos(x)
We can thus write sec^2(x) + 2 \tan^2(x) as
1/\cos^2(x) + 2 \sin^2(x)/\cos^2(x)= {1+2\sin^2(x)}/{\cos^2(x)}.
We want this expression to equal 4, so we can multiply for \cos^2(x) and get
{1+2\sin^2(x)}/{\cos^2(x)}=4 \iff 1+2\sin^2(x)=4\cos^2(x).
Subtracting \cos^2(x) from both sides, we have
1-\cos^2(x) +2\sin^2(x)=3\cos^2(x),
and since 1-\cos^2(x) equals \sin^2(x), the expression becomes
3\sin^2(x)=3\cos^2(x).
Dividing the whole expression by the right member, we have
{3\sin^2(x)}/{3\cos^2(x)}=1. Canceling the 3's out, and recalling again that \tan(x)=\sin(x)/\cos(x), we finally find out that the previous equation is the same as
\tan^2(x)=1, which is verified (considering only the angles between 0 and 2\pi) when \tan(x)=1, i.e. when x=\pi/4, and when \tan(x)=-1, i.e. when x={5\pi}/4