How to solve sin ^2 x - cos ^2 x=1/2

By Pythagorean identity cos^2(x) = 1 - sin^2(x)
problem can be rewritten as:

sin^2(x) - [1 - sin^2(x)] = 1/2
2sin^2(x) - 1 = 1/2

4sin^2(x) - 2 = 1

4sin^2(x) = 3

sin^2(x) = 3/4

sin(x) = ±√(3)/2

x = PI/3+ kPI or x = 2PI/3 + kPI

recall that cos(x)^2 - sin(x)^2 = cos(2x). Therefore sin(x)^2 - cos(x)^2 = -cos(2x)

-cos(2x) = 1/2
cos(2x) = -1/2
2x = 2pi/3 + 2 * pi * k , 4pi/3 + 2 * pi * k
x = pi/3 + pi * k , 2pi/3 + pi * k
x = (pi/3) * (1 + 3k) , (pi/3) * (2 + 3k)