Introducing false solutions - trig

Hi there
I'm having problems detecting when I'm introducing false solutions into my trig work.

I'm doing a question that states:

Given that h(x) = sin x +cos x, 0<x<2pi, find the values of x for which h'(x) = 0.

Here's my working. There are (according to the mark scheme), two correct and two false solutions in my answer.

h'(x) = cos x - sin x = 0
sin x = cos x
root(1-cos^2 x) = cos x
1- cos^2 x = cos^2 x
1 = 2 cos^2 x
1/2 =cos^2 x
+/- root2 / 2 = cos x

so x = pi/4 (correct), 7pi/4 (incorrect), 3pi/4 (incorrect), 5pi/4 (correct)

I know how to work out the answer the right way to get the correct solutions alone, but what's wrong with this way?

Extra solutions are often introduced when you square both sides.

Ahh! Nice to know. thanks!

also, prsom