Prove the following identities. Identities and Equations

Expert Answers
lfryerda eNotes educator| Certified Educator

Each part of your question is a different identity.

(a) `1/{cos\theta}+tan\theta={1+sin\theta}/cos\theta`





(b) `(sin x+cos x)^2=1+2sin x cos x`

`LS=(sin x + cos x)^2`

`=sin^2 x+2 sin x cos x +cos^2 x`

`=sin^2 x+cos^2 x + 2sin x cos x`

`=1+2 sin x cos x`


(c) `1+cos 2x=cot x sin 2x`

`RS=cot x sin 2x`

`={cos x}/{sin x} 2 sin x cos x`

`=2 cos^2 x`

`=1+2 cos^2 x - 1`

`=1+cos 2x`


(d) `tan^2x ={1-cos 2x}/{1+cos 2x}`

`RS={1-cos 2x}/{1+cos 2x}`

`={1-(1-2 sin^2 x)}/{1+2cos^2-1}`

`={2sin^2 x}/{2 cos^2 x}`

`={sin^2 x}/{cos^2 x}`

`=tan^2 x`


In each case the only identities that were used are the following standard ones, with some algebraic manipulation.

`sin^2 x+cos^2 x=1`

`sin2x=2 sin x cos x`

`cos 2x = 2 cos^2x-1`

`cos 2x=1-2sin^2x`

`tanx={sin x}/{cos x}`

`cot x = {cos x}/{sin x}`