Given: sin A + sin² A = 1

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

Now, consider cos²A + cos⁴A = cos² A(1 + cos²A)

Put the value of cos²A in the above equation:

Therefore, cos²A + cos⁴A = cos² A(1 + cos²A)

= (1 – sin² A)(1+1 – sin² A)

Again, from equation (1), we have 1 – sin² A = sin A. So, put the value of sin A in the above equation:

Therefore, cos²A + cos⁴A = (sinA)(1+ sinA)

= sin A + sin² A

= 1 (given)

Therefore, cos²A + scos⁴A = 1