Question

Solve each trig identity:

a) cos^4(x) - sin^4(x) = cos^2(x)

b) cos4(x) = cos^2(x) - sin^2(x)

Answer 1

Those are not identities. You solve equations; you prove identities.
(cos^2x + sin^2x)(cos^2x - sin^2x) = cos^2x
2cos^2x - 1 = cos^2(x)
cos^2x = 1
cosx = ±1
x = kπ for any integer k

Not sure what cos4(x) means. Is it
cos^4(x) or cos(4x) ??
If it's cos^4(x) then it's almost like the first one
cos^4(x) = 2cos^2(x) - 1
cos^4(x) - 2cos^2(x) + 1 = 0
(cos^2(x) - 1)^2 = 0
cos^2(x) = 1
x = kπ