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

We'll re-write the equation, substituting sin 2x by
2sinx*cosx.

We'll re-write now the entire
expression.

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

We'll factorize by cos x and we'll
get:

cos x * (cos x + 2sin x) =
0

We'll put each factor from the product as
0.

cos x = 0

This is an
elementary equation.

x = arccos 0 +
2k*pi

x = pi/2 +
2k*pi

or

x
= 3pi/2 + 2k*pi

cos x + 2sin x =
0

This is a homogeneous equation, in sin x and cos
x.

We'll divide the entire equation, by cos
x.

1 + 2 sinx/cos x = 0

But
the ratio sin x / cos x = tg x. We'll substitute the ratiosin x / cos x by tg
x.

1 + 2tan x= 0

We'll
subtract 1 both sides:

2tan x =
-1

We'll divide by 2:

tan x =
-1/2

x = arctg(-1/2 )
+k*pi

x = - arctg(1/2) +
k*pi

The solutions of the equation
are:

{pi/2 + 2k*pi}U{3pi/2 + 2k*pi}U{-
arctg(1/2) + k*pi}