This is an identity that can be easily proven using trigonometric properties.

Starting with the left side:
sin^2(x) + cos^2(x) + sin^2(3x)
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1:
1 + sin^2(3x)
Now, using the triple angle formula, sin(3x) = 3sin(x) - 4sin^3(x):
1 + (3sin(x) - 4sin^3(x))^2
Expanding and simplifying:
1 + 9sin^2(x) - 24sin^4(x) + 16sin^6(x)
Rearranging terms:
16sin^6(x) - 24sin^4(x) + 9sin^2(x) + 1

We can see that the expression we derived does not simplify to 3/2, therefore the given equation does not hold true.