2(sin^4 (x) + cos^4 (x)) = 1
sin^4 (x) + cos^4 (x) = 1/2
(sin^2 (x))^2 + cos^4 (x) = 1/2
(1 - cos^2 (x))^2 + cos^4 (x) = 1/2
1 - 2cos^2 (x) + cos^4 (x) + cos^4 (x) = 1/2
2cos^4 (x) - 2cos^2 (x) + 1/2 = 0
4cos^4 (x) - 4cos^2 (x) + 1 = 0
(2cos^2 (x) - 1)^2 = 0
2cos^2 (x) - 1 = 0
2cos^2 (x) = 1
cos^2 (x) = 1/2
cos (x) = 1 / sqrt(2) or cos (x) = -1 / sqrt(2)
Let us consider cos(x) = 1 / sqrt(2) on [-pi, pi] first.
In this case, x = pi/4 or x = -pi/4
Now, we will consider cos(x) = -1 / sqrt(2) on [-pi, pi].
In this case, x = -3pi/4 or x = 3pi/4
Hence, the solutions will be x = -pi/4, pi/4, -3pi/4 or 3pi/4