Cos2x Formula, Derivation & Solved Examples

cos2x formula is one of the most important trigonometric formulas for calculating the cosine function for the compound angle 2x. The trigonometric ratio of an angle in a right triangle describes the relationship between the angle and the lengths of its sides. One such trigonometric ratio is cos 2X,

The cos2x formula is as follows,

Cos 2x = Cos²x – Sin²x
= 2Cos²x– 1
= 1 – 2sin²x

The Cos2x formula can be derived using several trigonometric identities. In this article, we will look at the cos2x formula in terms of several trigonometric functions, as well as its derivation.

what is Cos2x?

There are several trigonometric identities, cos function being one of them. The Cosine of an angle is defined as the ratio of the side nearest to the angle to the hypotenuse (longest side) in a triangle. The question now is, Did you know what is the meaning of cos2x?
Cos2x is a proper trigonometric function for calculating the cosine function for the compound angle 2x. Cos2x is a trigonometric function for determining the value of cos when the angle x is doubled.

What is Cos2x formula?

Cos2x formula is a double-angle formula in trigonometry that is used to calculate the value of the Cosine Function for two angles. It is an important trigonometric identity that can be applied to a wide range of trigonometric and integration problems. The cos2x formula can be expressed using many trigonometric functions and formulas as follows

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

Derivation of Cos 2 Theta formula

Cos2x is a double-angle trigonometric identity because the angle under consideration is a multiple of 2, i.e. the double of x. The Cos2x Formula is a crucial identity in trigonometry that can be represented using trigonometric functions like sine, cosine, and tangent. Let’s look at the derivations of the Cos2x Formula.

Cos2x Formula Using Angle Addition Formula

The Cos2x Formula for the cosine function can be derived using the Angle Addition Formula. Angle 2x is denoted by the formula 2x = x + x. The Cos2x Formula can be simply demonstrated using the provided identities.

Cos (a + b) = cos a cos b – sin a sin b.

Substitute a = b = x into the formula for cos (a + b) using the angle addition formula for the sine function.

cos2x = cos (x + x)

= cos x cos x – sin x sin x

= cos²x – sin²x

This leads to the universal  Cos2x Formula cos2x = cos²x – sin²x is derived.

Cos2x Formula in terms of Sinx

Now that we have established that cos2x = cos²x – sin²x, we can calculate the cos2x formula solely in terms of the sine function. To demonstrate that cos2x = 1 – 2sin²x, we will use the trigonometry identity cos²x + sin²x = 1.

cos2x = cos²x – sin²x

= (1 – sin²x) – sin²x [Since cos²x + sin²x = 1 ⇒ cos²x = 1 – sin²x]

= 1 – sin²x – sin²x

= 1 – 2sin²x

So, cos2x formula in terms of sin x, cos2x = 1 – 2sin²x.

Cos2x Formula in terms of Cosx

In terms of Cosx, the Cos2x Formula is cos2x = 2cos²x – 1. The trigonometric identities cos2x = cos²x – sin²x and cos²x + sin²x = 1 can be used to calculate it.

cos2x = cos²x – sin²x

= cos²x – (1 – cos²x) [cos²x + sin²x = 1 ⇒ sin²x = 1 – cos²x]

= cos²x – 1 + cos²x

= 2cos²x – 1

Thus, in terms of Cos, the Cos2x Formula is Cos is cos2x = 2cos²x – 1.

Cos2x Formula in terms of Tanx

Various trigonometric identities and trigonometric formulas can be used to obtain the Cos2x Formula in terms of the Tangent Function. To derive the cos2x formula in terms of tan x, we will employ a few trigonometric identities and trigonometric formulas such as

• cos2x = cos²x – sin²x
• cos²x + sin²x = 1
• tan x = sin x/ cos x

cos2x = cos²x – sin²x

= (cos²x – sin²x)/1

= (cos²x – sin²x)/( cos²x + sin²x) [cos²x + sin²x = 1]

Divide cos2x by 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) [Because tan x = sin x / cos x]

Such that Cos2x Formula in terms of Tanx, is cos2x = (1 – tan2x)/(1 + tan2x).

1 Cos2x Formula

The formula for 1 $\cos (2x)$ is derived from trigonometric identities. It can be expressed using the double-angle formula for cosine:

1 cos(2x)=cos2(x)−sin2(x)=2cos2(x)−1=1−2sin2(x)

These formulas provide different ways to represent cos(2x) in terms of trigonometric functions of x. You can choose the form that best suits your needs in a particular context.

Sin2x Cos2x Formula

Sin2x Cos2x Formula is given below.

Cos2x Formula Examples

Example 1: Determine the  Cos2x Formula in terms of Cot x.

Solution: Earlier we came to know that,

cos2x = (1 – tan²x)/(1 + tan²x) and tan x = 1/cot x

cos2x = (1 – tan²x)/(1 + tan²x)

= (1 – 1/cot²x)/(1 + 1/cot²x)

= (cot²x – 1)/(cot²x + 1)

As a results, we can write the Cos2x Formula in terms of Cot x is cos2x = (cot²x – 1)/(cot²x + 1).

Example 2: Demonstrate the cosine function’s triple angle identity using the cos2x formula.

Solution:

The triple angle identity of the cosine function 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 [ 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) [As, cos2x + sin2x = 1 ⇒ sin2x = 1 – cos2x]

= 2 cos3x – cos x – 2 cos x + 2 cos3x

= 4 cos3x – 3 cos x.

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