Given: cos A + cos² A = 1

Therefore cos A = 1 – cos² A = sin² A ……(1)

Now, consider sin²A + sin⁴A = sin² A(1 + sin²A)

Put the value of sin²A in the above equation:

Therefore, sin²A + sin⁴A = sin² A(1 + sin²A)

= (1 – cos² A)(1+1 – cos² A)

Again, from equation (1), we have 1 – cos² A = cos A. So put the value of cos A in the above equation:

Therefore, sin²A + sin⁴A = (cosA)(1+ cosA)

= cos A + cos² A

= 1 (given)

Therefore, sin²A + sin⁴A = 1