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.

What is Cos2x?

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.

What is Cos2x Formula in Trigonometry?

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 = cos 2 x – sin 2 x
• cos2x = 2cos 2 x – 1
• cos2x = 1 – 2sin 2 x
• cos2x = (1 – tan 2 x)/(1 + tan 2 x)

Derivation of Cos2x Formula Using Angle Addition Formula

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

= cos 2 x – sin 2 x

Hence, we have cos2x = cos 2 x – sin 2 x

Cos2x In Terms of sin x

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

cos2x = cos 2 x – sin 2 x

= (1 – sin 2 x) – sin 2 x [Because cos 2 x + sin 2 x = 1 ⇒ cos 2 x = 1 – sin 2 x]

= 1 – sin 2 x – sin 2 x

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

Cos2x In Terms of cos x

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

cos2x = cos 2 x – sin 2 x

= cos 2 x – (1 – cos 2 x) [Because cos 2 x + sin 2 x = 1 ⇒ sin 2 x = 1 – cos 2 x]

= cos 2 x – 1 + cos 2 x

Hence , we have cos2x = 2cos 2 x – 1 in terms of cosx

Cos2x In Terms of tan x

Now, that we have derived cos2x = cos 2 x – sin 2 x, we will derive cos2x in terms of tan x. We will use a few trigonometric identities and trigonometric formulas such as cos2x = cos 2 x – sin 2 x, cos 2 x + sin 2 x = 1, and tan x = sin x/ cos x. We have,

cos2x = cos 2 x – sin 2 x

= (cos 2 x – sin 2 x)/1

= (cos 2 x – sin 2 x)/( cos 2 x + sin 2 x) [Because cos 2 x + sin 2 x = 1]

Divide the numerator and denominator of (cos 2 x – sin 2 x)/( cos 2 x + sin 2 x) by cos 2 x.

(cos 2 x – sin 2 x)/(cos 2 x + sin 2 x) = (cos 2 x/cos 2 x – sin 2 x/cos 2 x)/( cos 2 x/cos 2 x + sin 2 x/cos 2 x)

= (1 – tan 2 x)/(1 + tan 2 x) [Because tan x = sin x / cos x]

Hence, we have cos2x = (1 – tan 2 x)/(1 + tan 2 x) in terms of tan x

Cos^2x (Cos Square 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.

Cos^2x Formula

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 ⇒ cos 2 x = 1 – sin 2 x
• cos^2x = cos2x + sin^2x ⇒ cos 2 x = cos2x + sin 2 x
• cos^2x = (cos2x + 1)/2 ⇒ cos 2 x = (cos2x + 1)/2

How to Apply Cos2x Identity?

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 = cos 2 x – sin 2 x and sin 60° = √3/2, cos 60° = 1/2. Since 2x = 120°, x = 60°. Therefore, we have

cos 120° = cos 2 60° – sin 2 60°

Important Notes on Cos 2x

• cos2x = cos 2 x – sin 2 x
• cos2x = 2cos 2 x – 1
• cos2x = 1 – 2sin 2 x
• cos2x = (1 – tan 2 x)/(1 + tan 2 x)
• 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.

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Cos2x Examples

Example 1: Prove the triple angle identity of cosine function using cos2x formula. Solution: The triple angle identity of the cosine function is cos 3x = 4 cos 3 x – 3 cos x To begin with, we will use the angle addition formula of the cosine function. cos 3x = cos (2x + x) = cos2x cos x – sin 2x sin x = (2cos 2 x – 1) cos x – 2 sin x cos x sin x [Because cos2x = 2cos 2 x – 1 and sin2x = 2 sin x cos x] = 2 cos 3 x – cos x – 2 sin 2 x cos x = 2 cos 3 x – cos x – 2 cos x (1 – cos 2 x) [Because cos 2 x + sin 2 x = 1 ⇒ sin 2 x = 1 – cos 2 x] = 2 cos 3 x – cos x – 2 cos x + 2 cos 3 x = 4 cos 3 x – 3 cos x Answer: Hence, we have proved cos 3x = 4 cos 3 x – 3 cos x using the cos2x formula.

Example 2: Express the cos2x formula in terms of cot x. Solution: We know that cos2x = (1 – tan 2 x)/(1 + tan 2 x) and tan x = 1/cot x cos2x = (1 – tan 2 x)/(1 + tan 2 x) = (1 – 1/cot 2 x)/(1 + 1/cot 2 x) = (cot 2 x – 1)/(cot 2 x + 1) Answer: Hence, cos2x = (cot 2 x – 1)/(cot 2 x + 1) in terms of cotangent function.

Example 3: Determine the derivative and integral of cos2x. Solution: To find the derivative of cos2x, we will use the chain rule method. d(cos2x)/dx = d(cos2x)/d(2x) × d(2x)/dx = -sin 2x × 2 = -2 sin 2x To find the integral of cos2x, assume that 2x = u. Then 2 dx = du (or) dx = du/2. Substituting these values in the integral ∫ cos2x dx, ∫ cos2x dx = ∫ cos u (du/2) = (1/2) ∫ cos u du We know that the integral of cos x is sin x + C. So, (1/2) ∫ cos u du = (1/2) sin u + C = (1/2) sin2x + C Answer: Hence, integral of cos2x is (1/2) sin2x + C and its derivative of cos2x is -2 sin2x.

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