How do you prove tan (2x) = (2tan (x) ) / (1-tan2 x) ?

Given

tan(2x)

To prove

tan(2x) = (2 tan x) / (1 – tan²x)

Proof

First let us start from LHS.

tan(2x)

We know that tan x = sin x / cos x.

sin(2x) / cos(2x)

We know that sin 2A = 2 sin A cos A.

2 sin x cos x / cos(2x)

Also cos 2A = cos²A – sin²A.

2 sin x cos x / (cos²x – sin²x) =

Divide the numerator and denominator by cos²x.

(2 sin x cos x / cos²x) / [(cos²x – sin²x) / cos²x] =

[2 sin x(1) / cos x] / [(cos²x / cos²x) – (sin²x / cos²x)] =

[2(sin x / cos x)] / [1 – (sin²x / cos²x)] =

Now tan x = sin x / cos x.

(2 tan x) / [1 – (sin²x / cos²x)] =

Remember that tan²x = sin²x / cos²x.

(2 tan x) / (1 – tan²x)

=RHS

Hence Proved

Alternative method

LHS=tan{(2x)}

=tan(x+x)

We know that tan (A +B)= tan(A)+tan(B) / 1- tan A tan B

Applying the same formula we get

=>tan(x+x)= tan(x)+tan(x) / 1- tan x tan x

2tan(x) / (1 – tan²x)

= RHS

Hence Proved

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