Grade 12 math help?

Question

a) How to prove that (sin2x) / (1 + cos2x) = tanx

b) How to prove that (1 - cosx) / sinx = tan (x/2)

Please and thanks

Answer 1

a). The trigonometric equation is (sin 2x)/(1 + cos 2x) = tan x.

Left hand side identity : (sin 2x)/(1 + cos 2x).

Double angle formulas : sin(2x) = 2 sin x cos x,

                                      cos(2x) = 2cos² x - 1.

(sin 2x)/(1 + cos 2x) = [2 sin x cos x]/[1 + 2cos² x - 1]

= [2 sin x cos x]/2cos² x

= sin x/cos x

Trigonometric function : tan x = sin x/cos x.

= tan x

= Right hand side identity.

Hence proved.

b).

The trigonometric equation is (1 - cos x)/(sin x) = tan (x/2).

Left hand side identity : (1 - cos x)/(sin x).

Half angle formulas : sin(x) = 2 sin (x/2) cos (x/2),

                                cos(x) = 1 - 2sin² (x/2).

(1 - cos x)/(sin x) = [1 - (1 - 2sin² (x/2))]/[2 sin (x/2) cos (x/2)]

= [1 - 1 + 2sin² (x/2)]/[2 sin (x/2) cos (x/2)]

= [2sin² (x/2)]/[2 sin (x/2) cos (x/2)]

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

= tan (x/2)

= Right hand side identity.

Hence proved.