Tan 2A Formula – Proof and Examples

Here you will learn what is the formula of tan 2A in terms of tan with proof and examples based on it.

Let’s begin –

Tan 2A Formula :

The formula of tan 2A is $2 tan A\over 1 – tan^2 A$

Proof :

We have,

tan (A + B) = $tan A + tan B\over 1 – tan A tan B$

Replacing B by A,

$\implies$  tan 2A = $tan A + tan A\over 1 – tan A tan A$

$\implies$  tan 2A = $2 tan A\over 1 – tan^2 A$

We can also write above relation in terms of angle A/2, just replace A by A/2, we get

tan 2A = $2 tan ({A\over 2})\over 1 – tan^2 ({A\over 2})$

Example : Find the value of Tan 120 Degrees ?

Solution : We Know that tan 60 = $\sqrt{3}$.

By using above formula, tan 2A = $2 tan A\over 1 – tan^2 A$

tan 120 = $2 tan 60\over 1 – tan^2 60$ = $2 \times \sqrt{3}\over 1 – 3$

$\implies$  tan 120 = $-\sqrt{3}$

Example : If sin A = $3\over 5$, where 0 < A < 90 degrees, find the value of tan 2A ?

Solution : We have,

sin A = $3\over 5$ where 0 < A < 90 degrees

$\therefore$ $cos^2 A$ = 1 – $sin^2 A$

$\implies$ cos A = $\sqrt{1 – sin^2 A}$ = $\sqrt{1 – {9\over 25}}$ = $4\over 5$

$\implies$ tan A = $sin A\over cos A$ = $3/5\over 4/5$ = $3\over 4$

By using above formula,

tan 2A = $2 tan A\over 1 – tan^2 A$ = $2 \times {3\over 4} \over 1 – {9\over 16}$

$\implies$ tan 2A = ${6\over 4}\over {7\over 16}$

$\implies$  tan 2A = $24\over 7$