Question

Prove the identity:

√[(1 – cos 2x)/(1 + cos 2x)] = tan x

Answer 1

Let us consider the LHS

√[(1 – cos 2x)/(1 + cos 2x)]

As we know that cos 2x = 1 – 2 sin² x

= 2 cos² x – 1

Therefore,

√[(1 – cos 2x)/(1 + cos 2x)] = √[(1 – (1 – 2sin² x))/(1 + (2cos²x – 1))]

= √[(1 – 1 + 2sin² x)/(1 + 2cos² x – 1)]

= √[2 sin² x/2 cos² x]

= sin x/cos x

= tan x

= RHS

Thus proved.