**Cos2x Formula**

The value of the cosine function, which is a trigonometric function, may be determined in trigonometry by using the cos2x identity, which is one of the main trigonometric identities.

The cosine function may also be referred to as a double angle identity in certain contexts. Because of the identity of cos2x, it is possible to describe the cosine of a compound angle 2x in terms of the sine and cosine trigonometric functions, in terms of the cosine function alone, in terms of the sine function only, and in terms of the tangent function solely.

The cosine function for the compound angle 2x may be found by using an essential trigonometric function called cos2x. This function finds the value of the cosine function. Cos2x may be expressed in terms of a variety of trigonometric functions, and each of its formulae can be used to simplify complicated trigonometric equations and solve issues involving integration. The value of cos is determined using the cos2x function, which is a double angle trigonometric function. This function is used to find the value of cos when the angle x is doubled.

**Cos2x Formula in trigonometry**

In trigonometry, the identity cos2x plays a significant role and may be stated in a variety of different ways. It is possible to represent it in terms of a variety of trigonometric functions, including sine, cosine, and tangent, among others. Since the angle under examination is a factor of 2, or the double of x, the cosine of 2x is an identity that belongs to the category of double angle trigonometric identities. Let us write the identity of cos2x using a few alternative forms:

- cos2x = cos2x – sin2x
- cos2x = 2cos2x – 1
- cos2x = 1 – 2sin2x
- cos2x = (1 – tan2x)/(1 + tan2x)

**Solved Examples**

Example 1: Using the cos2x formula, demonstrate the triple angle identity of the cosine function.

Solution:

cosine function’s triple angle identity is cos 3x = 4 cos3x – 3 cos x

cos 3x = cos (2x + x) = cos2x cos x – sin 2x sin x

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

= 2 cos3x – cos x – 2 sin2x cos x

= 2 cos3x – cos x – 2 cos x (1 – cos2x) [Since cos2x + sin2x = 1 ⇒ sin2x = 1 – cos2x]

= 2 cos3x – cos x – 2 cos x + 2 cos3x

= 4 cos3x – 3 cos x.