Question

If cos⁴ x/cos² y + sin⁴x/sin²y = 1, then prove that cos⁴y/cos² x + sin⁴y/sin² x = 1

Answer 1

The given condition is

cos⁴x sin²y + sin⁴x cos²y = sin²ycos²y

= sin²y(1 −sin²y) = sin²y −sin⁴y

Therefore,

Similarly, on the R.H.S. of Eq. (1), replacing sin²y by 1 − cos²y and simplifying as shown above, we get

cos⁴y/cos²x = cos² y + cos² y sin²x - cos² x sin²y

By adding Eqs. (2) and (3), we get the desired result.