Prove that sin^4 x+cos^4 x+(sin^2 2x)/2=1

We'll use the Pythagorean identity to solve the
problem.

(sin x)^2 + (cos x)^2 =
1

If we'll raise to square both sides, we'll
get:

[(sin x)^2 + (cos x)^2]^2 =
1^2

(sin x)^4 + (cos x)^4 + 2 (sin x)^2 *(cos x)^2 =
1

We'll keep the sum (sin x)^4 + (cos x)^4 to the
left:

(sin x)^4 + (cos x)^4 = 1 - 2 (sin x)^2 *(cos
x)^2

We'll also apply the double angle
identity:

sin 2x = 2 sin x*cos
x

We'll raise to square both
sides:

(sin 2x)^2 = 4 (sin x)^2 *(cos
x)^2

We'll divide by 2:

[(sin
2x)^2]/2 =2 (sin x)^2 *(cos x)^2

We'll re-write the
identity to be proved:

1 - 2 (sin x)^2 *(cos x)^2 + 2 (sin
x)^2 *(cos x)^2 = 1

We'll eliminate like
terms:

1 =
1

We'll get equal values both sides,
therefore the identity (sin x)^4 + (cos x)^4 + [(sin 2x)^2]/2 = 1 is
verified.