cos^4x ÷ cos^2y + sin^4x ÷ sin^2y

= cos^4(x) sec^2(y) + sin^4(x) csc^2(y)

cos(x)^4 / cos(y)^2 + sin(x)^4 / sin(y)^2 =>

(cos(x)^4 * sin(y)^2 + cos(y)^2 * sin(x)^4) / (sin(y)^2 * cos(y)^2) =>

(cos(x)^4 * (1 - cos(y)^2) + cos(y)^2 * sin(x)^4) / (sin(y)^2 * cos(y)^2) =>

(cos(x)^4 - cos(x)^4 * cos(y)^2 + cos(y)^2 * sin(x)^4) / (sin(y)^2 * cos(y)^2) =>

(cos(x)^4 + cos(y)^2 * (sin(x)^4 - cos(x)^4)) / (sin(y)^2 * cos(y)^2) =>

(cos(x)^4 + cos(y)^2 * (sin(x)^2 - cos(x)^2) * (sin(x)^2 + cos(x)^2)) / (sin(y)^2 * cos(y)^2) =>

(cos(x)^4 + cos(y)^2 * (-cos(2x)) * 1) / (sin(y)^2 * cos(y)^2) =>

(cos(x)^4 - cos(y)^2 * cos(2x)) / ((1/2)^2 * 4 * (sin(y) * cos(y))^2) =>

4 * (cos(x)^4 - cos(y)^2 * cos(2x)) / (2 * sin(y) * cos(y))^2 =>

4 * (cos(x)^4 - cos(y)^2 * cos(2x)) / sin(2y)^2

4 * (cos(x)^4 - cos(y)^2 * cos(2x)) / sin(2y)^2 = 1

4 * (cos(x)^4 - cos(y)^2 * cos(2x)) = sin(2y)^2

4 * cos(x)^4 - 4 * cos(y)^2 * cos(2x) = sin(2y)^2

4 * cos(x)^4 = 4 * cos(y)^2 * cos(2x) + sin(2y)^2

4 * cos(x)^4 = 4 * cos(y)^2 * cos(2x) + 4 * sin(y)^2 * cos(y)^2

cos(x)^4 = cos(y)^2 * cos(2x) + sin(y)^2 * cos(y)^2

cos(x)^4 = cos(y)^2 * cos(2x) + (1 - cos(y)^2) * cos(y)^2

cos(x)^4 = cos(y)^2 * cos(2x) + cos(y)^2 - cos(y)^4

cos(y)^4 - cos(y)^2 * (1 + cos(2x)) + cos(x)^4 = 0

cos(y)^4 - cos(y)^2 * (1 + cos(x)^2 - sin(x)^2) + cos(x)^4 = 0

cos(y)^4 - cos(y)^2 * (2 * cos(x)^2) + cos(x)^4 = 0

cos(y)^4 - 2 * cos(x)^2 * cos(y)^2 + cos(x)^4 = 0

cos(y)^2 = u

u^2 - 2 * cos(x)^2 * u + cos(x)^4 = 0

u = (2 * cos(x)^2 +/- sqrt(4 * cos(x)^4 - 4 * cos(x)^4)) / 2

u = 2 * cos(x)^2 / 2

u = cos(x)^2

cos(y)^2 = cos(x)^2

cos(y) = +/- cos(x)

cos(y) = cos(x)

cos(y) = cos(x + 2pi * k)

y = x + 2pi * k

cos(y) = -cos(x)

cos(y) = cos(pi - y + 2pi * k)

y = pi - y + 2pi * k

2y = pi + 2pi * k

y = pi/2 + pi * k

y = x + 2pi * k , pi/2 + pi * k

k is an integer.

So this only works when y = x (or some value of x that is coterminal to y) or when y = pi/2. Test

cos(x)^4 / cos(y)^2 + sin(x)^4 / sin(y)^2 =>

cos(x)^4 / cos(x)^2 + sin(x)^4 / sin(x)^2 =>

cos(x)^2 + sin(x)^2 =>

1

cos(x)^4 / cos(y)^2 + sin(x)^4 / sin(y)^2 =>

cos(x)^4 / cos(pi/2)^2 + sin(x)^4 / sin(pi/2)^2 =>

cos(x)^4 / 0^2 + sin(x)^4 / 1^2

So y = pi/2 + pi * k is extraneous

y = x is the only time when this works.

We can't, because it isn't true for all values of x and y.

or

prove

(cos^4x)(sin^2y)+(cos^2y)(sin^4x)=(sin^2y)(cos^2y)

Hint: sin^2(z)+cos^2(z)=1