Question

cosx / 1+sinx = 1-sinx / cosx
how can I prove this ?

Answer 1

LHS
= cos(x)/(1-sin(x)) + (1-sin(x))/cos(x)
= (cos²(x) + (1 - sin(x))²) / [cos(x)(1-sin(x)]
= (cos²(x) + sin²(x) - 2sin(x) + 1) / [cos(x)(1-sin(x)]
= (1 + 1 - 2sin(x)) / [cos(x)(1-sin(x)]
= 2(1 - sin(x)) / [cos(x)(1-sin(x)]
= 2/cos(x)
= 2sec(x)
= RHS

Answer 2

So start with the left side, try to match it to the right.
\[\large \frac{\cos x}{1+\sin x}\color{royalblue}{\left(\frac{1-\sin x}{1-\sin x}\right)}\]This would be your first step. Multiply by this blue fraction.

Answer 3

If the problem says "prove", you can't just cross multiply. You're trying to manipulate one side to match the other.