This is impossible because it isn't true.
Let me give a counter example that disproves the statement:
If you solve sin(θ)+sin2(θ) = 1, one of the many possible answers is 38.173°.
If you plug that angle into the second equation, you get:
cos2(38.173)+sin4(38.173) = 0.764 ≠ 1
Therefore the statement is false.
Are you certain the original question isn't this?
If sinθ+sin2θ=1, prove that cos2θ+cos4θ=1. This is actually provable.