# Tan2x Formula

Sin 2x, Cos 2x, Tan 2x are the trigonometric formulas known as double angle formulas because, they have double angles in their trigonometric functions. Let’s understand it by practising it through the solved examples.

 $$\begin{array}{l}Tan 2x =\frac{2tan x}{1-tan^{2}x} \end{array}$$

In terms of sine and cosine function, the tan 2x formula is written as follows:

 Tan 2x = sin 2x / cos 2x (or) Tan 2x = (2 sin x cos x) /(cos2 x – sin2 x)

## Tan 2x Formula Proof

Now, let’s look at the tan 2x trigonometric formulae derivation in detail.

With the help of sine and  cosine function, tan 2x formula can be derived as follows:

We know that tan x = sin x/cos x

Hence, tan 2x = sin 2x / cos 2x …(1)

Using sine and cosine double angle formula,

Sin 2x = 2 sin x cos x

Cos 2x = cos2 x – sin2x

Now, substitute the above two formulas in (1), we get

Tan 2x = (2 sin x cos x) /(cos2 x – sin2 x)

Now divide both numerator and denominator of  (2 sin x cos x) /(cos2 x – sin2 x) by cos2 x, we get

Tan 2x = [(2 sin x cos x)/cos2x ]/[(cos2 x – sin2 x)/cos2x]

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

Since (sin x/ cos x) = tan x and  (sin2x / cos2x) = tan2x, the above equation can be written as:

Tan 2x = 2 tan x / (1-tan2x)

Hence, the tan 2x formula can be derived with the help of sine and cosine functions.

### Examples on Tan 2x Formula

Example 1:

Find the value of tan 2x, if tan x = 5.

Solution:

Given: Tan x = 5

To find: Tan 2x

We know that the formula for tan 2x is:

Tan 2x = 2 tan x / (1-tan2x)

Now, substitute the known values in the formula, we get

Tan 2x = 2(5) / (1-(5)2)

Tan 2x = 10/ (1-25)

Tan 2x = 10/-24

Tan 2x = -5/12.

Hence, the value fo tan 2x is -5/12, if tan x = 5.

Example 2:

Find the value of tan 2x, if sin x = 5/12 and cos x = 3/12.

Solution:

Given: sin x = 5/12 and cos x = 3/12.

So, we have sin2x = 25/144, and cos 2x = 9/144

The formula for tan 2x, in terms of sine and cosine functions is:

Tan 2x = Sin 2x / Cos 2x

Tan 2x = (2 sin x cos x) /(cos2 x – sin2 x)

Now, substitute the known values in the formula, we get

Tan 2x = [2(5/12)(3/12)] / [(9/144) – (25/144)]

Tan 2x = (30/144)/(-16/144)

Tan 2x = -30/16, which on simplification, we get

Tan 2x = -15/8.

Hence, the value of tan 2x is -15/8.