If CosA+Cos2A=1,prove that Sin2A+Cos4A=1

If your question is like this:

For cos A + cos² A = 1, prove that sin² A + sin⁴ A = 1

Solution:

Given: cos A + cos² A = 1 .... (1)

⇒ cos A = 1 - cos² A = sin² A ... (2)

Now, sin² A + sin⁴ A = sin² A + (sin² A)²

= cos A + cos² A [using (2)]

= 1 [using (1)]

Hence proved