Cos2x

Cos2x is one of the important trigonometric identities used in trigonometry to find the value of the cosine trigonometric function for double angles. It is also called a double angle identity of the cosine function. The identity of cos2x helps in representing the cosine of a compound angle 2x in terms of sine and cosine trigonometric functions, in terms of cosine function only, in terms of sine function only, and in terms of tangent function only.

Cos2x identity can be derived using different trigonometric identities. Let us understand the cos2x formula in terms of different trigonometric functions and its derivation in detail in the following sections. Also, we will explore the concept of cos^2x (cos square x) and its formula in this article.

Cos2x is an important trigonometric function that is used to find the value of the cosine function for the compound angle 2x. We can express cos2x in terms of different trigonometric functions and each of its formulas is used to simplify complex trigonometric expressions and solve integration problems. Cos2x is a double angle trigonometric function that determines the value of cos when the angle x is doubled.

Cos2x is an important identity in trigonometry which can be expressed in different ways. It can be expressed in terms of different trigonometric functions such as sine, cosine, and tangent. Cos2x is one of the double angle trigonometric identities as the angle in consideration is a multiple of 2, that is, the double of x. Let us write the cos2x identity in different forms:

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

We know that the cos2x formula can be expressed in four different forms. We will use the angle addition formula for the cosine function to derive the cos2x identity. Note that the angle 2x can be written as 2x = x + x. Also, we know that cos (a + b) = cos a cos b - sin a sin b. We will use this to prove the identity for cos2x. Using the angle addition formula for cosine function, substitute a = b = x into the formula for cos (a + b).

cos2x = cos (x + x)

= cos x cos x - sin x sin x

= cos²x - sin²x

Hence, we have cos2x = cos²x - sin²x

Now, that we have derived cos2x = cos²x - sin²x, we will derive the formula for cos2x in terms of sine function only. We will use the trigonometry identity cos²x + sin²x = 1 to prove that cos2x = 1 - 2sin²x. We have,

cos2x = cos²x - sin²x

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

= 1 - sin²x - sin²x

= 1 - 2sin²x

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

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

cos2x = cos²x - sin²x

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

= cos²x - 1 + cos²x

= 2cos²x - 1

Hence , we have cos2x = 2cos²x - 1 in terms of cosx

Now, that we have derived cos2x = cos²x - sin²x, we will derive cos2x in terms of tan x. We will use a few trigonometric identities and trigonometric formulas such as cos2x = cos²x - sin²x, cos²x + sin²x = 1, and tan x = sin x/ cos x. We have,

cos2x = cos²x - sin²x

= (cos²x - sin²x)/1

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

Divide 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]

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

Cos^2x is a trigonometric function that implies cos x whole squared. Cos square x can be expressed in different forms in terms of different trigonometric functions such as cosine function, and the sine function. We will use different trigonometric formulas and identities to derive the formulas of cos^2x. In the next section, let us go through the formulas of cos^2x and their proofs.

To arrive at the formulas of cos^2x, we will use various trigonometric formulas. The first formula that we will use is sin^2x + cos^2x = 1 (Pythagorean identity). Using this formula, subtract sin^2x from both sides of the equation, we have sin^2x + cos^2x -sin^2x = 1 -sin^2x which implies cos^2x = 1 - sin^2x. Two trigonometric formulas that includes cos^2x are cos2x formulas given by cos2x = cos^2x - sin^2x and cos2x = 2cos^2x - 1. Using these formulas, we have cos^2x = cos2x + sin^2x and cos^2x = (cos2x + 1)/2. Therefore, the formulas of cos^2x 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

Cos2x formula can be used for solving different math problems. Let us consider an example to understand the application of cos2x formula. We will determine the value of cos 120° using the cos2x identity. We know 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

Important Notes on Cos 2x

• cos2x = cos²x - sin²x
• cos2x = 2cos²x - 1
• cos2x = 1 - 2sin²x
• cos2x = (1 - tan²x)/(1 + tan²x)
• The formula for cos^2x that is commonly used in integration problems is cos^2x = (cos2x + 1)/2.
• The derivative of cos2x is -2 sin 2x and the integral of cos2x is (1/2) sin 2x + C.

☛ Related Articles:

• Trigonometric Ratios
• Trigonometric Table
• Sin2x Formula
• Inverse Trigonometric Ratios

What is Cos2x Identity in Trigonometry?

Cos2x is one of the double angle trigonometric identities as the angle in consideration is a multiple of 2, that is, the double of x. It can be expressed in terms of different trigonometric functions such as sine, cosine, and tangent.

What is the Cos2x Formula?

Cos2x can be expressed in terms of different trigonometric functions such as sine, cosine, and tangent. It can be expressed as:

• cos2x = cos²x - sin²x
• cos2x = 2cos²x - 1
• cos2x = 1 - 2sin²x

What is the Derivative of cos2x?

The derivative of cos2x is -2 sin 2x. Derivative of cos2x can easilty be calculated using the formula d[cos(ax + b)]/dx = -asin(ax + b)

What is the Integral of cos2x?

The integral of cos2x can be easilty obtained using the formula ∫cos(ax + b) dx = (1/a) sin(ax + b) + C. Therefore, the integral of cos2x is given by ∫cos 2x dx = (1/2) sin 2x + C.

What is Cos2x In Terms of sin x?

We can express the cos2x formula in terms of sinx. The formula is given by cos2x = 1 - 2sin²x in terms of sin x.

What is Cos2x In Terms of tan x?

We can express the cos2x formula in terms of tanx. The formula is given by cos2x = (1 - tan²x)/(1 + tan²x) in terms of tan x.

How to Derive cos2x Identity?

Cos2x identity can be derived using different identities such as angle sum identity of cosine function, cos²x + sin²x = 1, tan x = sin x/ cos x, etc.

How to Derive Cos Square x Formula?

We can derive the cos square x formula using various trigonometric formulas which consist of cos^2x. The trigonometric identities which include cos^2x are cos^2x + sin^2x = 1, cos2x = cos^2x - sin^2x and cos2x = 2cos^2x - 1. We can simplify these formulas and determine the value of cos square x.

What is Cos^2x Formula?

We have three formulas for cos^2x given below:

• 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

What is the Formula of Cos2x in Terms of Cos?

The formula of cos2x in terms of cos is given by, cos2x = 2cos^2x - 1, that is, cos2x = 2cos²x - 1.