# 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 = $$24\over 7$$