# Prove the identity: (cosx-sinx)/(cosx+sinx) = sec(2x) - tan(2x)

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We have to prove: (cos x - sin x)/(cos x + sin x) = sec(2x) - tan(2x)

Start from the left hand side

`(cos x - sin x)/(cos x + sin x) `

=> `((cos x - sin x)(cos x - sin x))/((cos x + sin x)(cos x - sin x))`

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

=> `(1 - sin(2x))/(cos (2x))`

=> `1/(cos(2x)) - (sin 2x)/(cos 2x)`

=> `sec 2x - tan 2x`

**This proves that (cos x -sin x)/(cos x+sin x) = sec(2x) - tan(2x)**

R:H:S ≡ sec 2x - tan 2x

= (1 - sin 2x)/cos 2x

= (1-2sinx.cosx)/ (cos²x - sin²x)

= (sin²x + cos²x -2sinx.cosx) / (cosx - sinx)(cosx + sinx)

= (cosx - sinx)²/ (cosx - sinx)(cosx + sinx)

= (cosx - sinx) / (cosx + sinx)

= L:H:S