Find value of

Question

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

Answer 1

I sure don't see anything obvious, but
(2sinx cosx/cosx)^2 = 4sin^2x
(2sin(2x)cos(2x)/cos(2x))^2 = 4sin^2(2x) = 16sin^2x cos^2x
sin^2(x)/(cos^2(2x)-sin^2(2x))^2 = sin^2x/(sin^4x+cos^4x-6sin^2x cos^2x)
adding those up, you get
4sin^2x + 16sin^2x cos^2x + sin^2x/cos^2(4x)
= sin^2(x)/cos^2(4x) (4cos(2x)+12cos(4x)+4cos(6x)+1)

there are so many ways to massage this kind of stuff, you can stop almost anywhere and say that's it, unless you have some specific target in mind.