Proving sec(2x) - tan(2x) and cos(2x)/(1 + sin(2x)) identities

[Hockeystar]

Homework Statement

Teacher gave this identity twice to us but the starting position slightly varies.

Prove sec(2x) - tan(2x) = (cos(x)-sin(x))/(cos(x) + sin(x))

Prove cos(2x)/(1 + sin(2x)) = tan(pi/4-x)

Homework Equations

pretty much all trig identities

The Attempt at a Solution

I get to (1-sin(2x))/cos(2x) = (cos(x)-sin(x))/(cos(x) + sin(x)) and then I'm stumped.

 

[tiny-tim]

Welcome to PF!

Hi Hockeystar! Welcome to PF!

I get to (1-sin(2x))/cos(2x) = (cos(x)-sin(x))/(cos(x) + sin(x))

ok so far …

now what is cos(2x) ? ​

 

[Hockeystar]

cosx - sinx(cosx + sinx). How does that help me cancel out something?

 

[tiny-tim]

cosx - sinx(cosx + sinx). How does that help me cancel out something?

You mean (cosx - sinx)(cosx + sinx) …

It doesn't have to cancel to help

think! look at the RHS!