# 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. Letâ€™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, letâ€™s look at the tan 2xÂ trigonometric formulaeÂ derivation in detail.

With the help of sine andÂ  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 â€¦(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) /(cos2Â x – sin2Â x)

Now divide both numerator and denominator ofÂ  (2 sin x cos x) /(cos2 x – sin2 x) by cos2Â x, we get

Tan 2x = [(2 sin x cos x)/cos2x ]/[(cos2Â x – sin2Â x)/cos2x]

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

Since (sin x/ cos x) = tan x andÂ  (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) /(cos2Â x – sin2Â 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.