Cos2x Formula: Derivation, Application and Question Example with Solution

Cos2X Formula is one of the essential trigonometric identities used to determine the value of the cosine trigonometric function for double angles.

Cos2x Formula in trigonometry can be expressed in terms of different trigonometric functions such as sine, cosine, and tangent. It is also known as the double angle identity of the cosine function. The identity of cos2x helps represent the cosine of a compound angle 2x in terms of sine and also the cosine trigonometric functions in terms of cosine function only, sine function only, and tangent function only.

Cos2x Formula in Trigonometry

The identity of the cos2x formula in trigonometry can be stated in various ways. Cos2x is represented in a variety of trigonometric functions, including sine, cosine, and tangent, among others. The cos2x formula belongs to the category of double angle trigonometric identities as the angle under examination is a factor of 2 or the double of x. The identity of cos2x in a few alternative forms is shown below:

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

What is Cos2x?

Also called a double angle identity of the cosine function, Cos2x is one of the many critical trigonometric identities used to find the value of the cosine trigonometric function for double angles. Cos2x is expressed in terms of different trigonometric functions and each of its formulas is used to simplify complex trigonometric expressions and solve integration problems. It is a double-angle trigonometric function that helps in finding out the value of cos when the angle x is doubled.

Derivation of the Cos2x Formula

The cos2x formula can be expressed in four different forms. The cosine value of a compound angle ‘2x’ is represented in terms of sine function only, in terms of cosine function only, in terms of sine and cosine trigonometric functions, and in terms of tangent function only. Shown below are some of the ways in which the Cos2x formula is derived:

Derivation of Cos2x Formula Using Angle Addition Formula

The Cos2x formula can be derived using the angle addition formula for the cosine function. The angle 2x can also be written as 2x = x + x. Also, we already know that cos (a + b) = cos a x cos b - sin a x sin b. This can be used to prove the identity of cos2x. Using the angle addition formula for the cosine function, we can substitute a = x and b = x in the formula for cos (a + b).

cos2x = cos (x + x)

= cos x x cos x - sin x x sin x

= cos2x - sin2x

Thus we have cos2x = cos2x - sin2x

Derivation of Cos2x Formula In Terms of Sin x

Now, that we have established cos2x = cos2x - sin2x, we will derive the formula for cos2x in terms of only the sine function. We can use the trigonometry identity cos2x + sin2x = 1 to derive the cos2x formula in terms of sin x. We have,

cos2x = cos2x - sin2x

= (1 - sin2x) - sin2x [As cos2x + sin2x = 1 ⇒ cos2x = 1 - sin2x]

= 1 - sin2x - sin2x

= 1 - 2sin2x

Therefore, in terms of sin x, we have cos2x = 1 - 2sin2x.

Derivation of Cos2x Formula In Terms of Cos x

Just like we have derived cos2x = 1 - 2sin2x, we will derive cos2x in terms of cos x, i.e. cos2x = 2cos2x - 1. Using the trigonometry identities cos2x = cos2x - sin2x and cos2x + sin2x = 1 to prove that cos2x = 2cos2x - 1, we have,

cos2x = cos2x - sin2x

= cos2x - (1 - cos2x) [As cos2x + sin2x = 1 ⇒ sin2x = 1 - cos2x]

= cos2x - 1 + cos2x

= 2cos2x - 1

Therefore, in terms of cos x, we have cos2x = 2cos2x - 1.

Derivation of Cos2x Formula In Terms of Tan x

Using the angle addition formula, we have derived cos2x = cos2x - sin2x. Now we will derive cos2x in terms of tan x using a few trigonometric identities and trigonometric formulas such as cos2x = cos2x - sin2x, cos2x + sin2x = 1, and tan x = sin x/ cos x.

We already know that,

cos2x = cos2x - sin2x

= (cos2x - sin2x)/1

= (cos2x - sin2x)/(cos2x + sin2x) [As cos2x + sin2x = 1]

Dividing the numerator and denominator of (cos2x - sin2x)/( cos2x + sin2x) by cos2x.

(cos2x - sin2x)/(cos2x + sin2x) = (cos2x/cos2x - sin2x/cos2x)/( cos2x/cos2x + sin2x/cos2x)

= (1 - tan2x)/(1 + tan2x) [Since tan x = sin x / cos x]

Therefore, in terms of tan x, we have cos2x = (1 - tan2x)/(1 + tan2x)

What is Cos^2x?

Cos^2x is a trigonometric function that indicates cos x whole squared. Cos square x can be expressed in different forms in terms of different trigonometric functions such as the cosine function, and the sine function. Various trigonometric formulas and identities can be used to derive the formulas of cos^2x as shown below.

Cos^2x Formula

Various trigonometric formulas are used to arrive at the formulas of cos^2x. Firstly, let us consider the formula sin^2x + cos^2x = 1 (Pythagorean identity). We will now subtract sin^2x from both sides of the equation to get sin^2x + cos^2x -sin^2x = 1 -sin^2x. This implies cos^2x = 1 - sin^2x.

The two trigonometric formulas that comprise cos^2x are cos2x = cos^2x - sin^2x and cos2x = 2cos^2x - 1. Using these two formulas, we get cos^2x = cos2x + sin^2x and cos^2x = (cos2x + 1)/2. Hence, the cos^2x formulas are:

cos^2x = 1 - sin^2x

⇒ cos2x = 1 - sin2x

cos^2x = cos2x + sin^2x

⇒ cos2x = cos2x + sin2x

cos^2x = (cos2x + 1)/2

⇒ cos2x = (cos2x + 1)/2

How to Apply the Cos2x Identity?

The cos2x formula is used for solving different math problems. Let us consider an example to demonstrate the application of the cos2x formula.

For Eg: We will determine the value of cos 120° using the cos2x identity. It is already known that cos2x = cos2x - sin2x and sin 60° = √3/2, cos 60° = 1/2. Since 2x = 120°, x = 60°. Therefore, we have

cos 120° = cos260° - sin260°

= (1/2)2 - (√3/2)2

= 1/4 - 3/4

= -1/2