Question: find x: (sin^2 x)(tan x) = 2 - cos^2 x
sin^2(x).tan(x) = 2 - cos^2(x) -- divide both sides by cos^2(x)
tan^2(x).tan(x) = 2.sec^2(x) - 1 -- use the identity 1 + tan^2 = sec^2
tan^3(x) = 2 + 2.tan^2(x) - 1 -- substitute u = tan(x)
u^3 = 2u^2 + 1
u^3 - 2u^2 - 1 = 0
You can use the Newton-Raphson method to solve for u.
I get u = 2.20557
u = tan(x)
Therefore x = 1.14512 rad