To solve the equation sinx + sin3x + cosx + cos3x = 0, we can use trigonometric identities to simplify the expression.

Using the sum-to-product identities:
sin3x = 2sinxcos2x
cos3x = 2cosxcos2x

Substitute these identities into the original equation:
sinx + 2sinxcos2x + cosx + 2cosxcos2x = 0

Factor out sinx and cosx:
sinx(1 + 2cos2x) + cosx(1 + 2cos2x) = 0

Now, we have a common factor of (1 + 2cos2x):
(1 + 2cos2x)(sinx + cosx) = 0

Set each factor to zero:
1 + 2cos2x = 0
sinx + cosx = 0

To solve the first equation:
2cos2x = -1
cos2x = -1/2
2x = 2π/3, 4π/3

To solve the second equation:
sinx = -cosx
tanx = -1
x = 3π/4

Therefore, the solutions to the equation sinx + sin3x + cosx + cos3x = 0 are x = 3π/4, 2π/3, and 4π/3.