Proving trigonomic expressions

Question

I need help solving this :

1. Prove the following trigonometric identities. (Neatly show all your work/proof steps.) [2x5=10 marks]

a) [(sin 2x) / (1+ cos 2x)] . [cos x / (1+ cos x)] = tan (x/2)

b) (sec2 2x – tan2 2x) / (sec 2x + tan 2x) = cos 2x / (1+ sin 2x)

Answer 1

a) as we have 1 + cos 2x and 1 + cos 2x, the easiest way to simplify is to make use of the identity

cos 2A = 2 cos² A - 1, as 1 + cos 2A = 2 cos² A and sin 2x = 2 sin x cos x

Then,

sin 2x/(1 + cos 2x) * cos x/(1 + cos x) =

2 sin x cos x/(2 cos²x) * cos x/(2 cos² x/2) =

sin x/(2 cos² x/2) =

2 sin x/2 cos x/2 /(2 cos² x/2) =

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

tan x/2

b)sec² - tan² = 1. Thus,

(sec² 2x- tan² 2x)/(sec 2x + tan 2x) =

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

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

cos 2x/(1 + sin 2x)