Question

Prove each of the following identities (a) sec x + tan x =cos x/1 − sin x (b) tan^2 x/tan^2 x + 1= sin^2 x?

Answer 1

a)

`LHS=sec x + tan x`

`=1/cosx + sinx/cosx`

`=((1 + sinx)cosx)/cos^2x`

`=((1 + sinx)cosx)/(1-sin^2x)`

`=((1 + sinx)cosx)/((1-sinx)(1+sinx))`

`=cos x/(1 − sin x)=RHS`

(b) `LHS=tan^2 x/(tan^2 x + 1)`

`=tan^2 x/sec^2 x`

`=(sin^2 x/cos^2x)/(1/cos^2 x`

`= sin^2 x=RHS`