(cos x)^2 + sin 2x = 0

(cosx)^2 + sin2x
= 0

We know that:

sin2x = 2sinx*cosx

==> cos^2 x + 2sinx*cosx = 0

Facros cos(x):

==> cosx*(cosx + 2sinx) = 0

Then :

cosx = 0  
OR   cosx + 2sinx = 0

cosx = 0 ==> x=pi/2 ,
3pi/2

cosx + 2sinx = 0

==> cosx =
-2sinx

==> -cosx/2 = sinx

Divide by cosx

==> -1/2 = tanx

==> tanx = -1/2

==> x= -arcttan (1/2)