What Does Cos 2x Equal

Cos2x

Cos2x is ane of the important trigonometric identities used in trigonometry to find the value of the cosine trigonometric office 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 simply, in terms of sine function only, and in terms of tangent part only.

Cos2x identity can be derived using dissimilar trigonometric identities. Let u.s.a. understand the cos2x formula in terms of different trigonometric functions and its derivation in detail in the following sections. Likewise, we will explore the concept of cos^2x (cos square 10) and its formula in this article.

1.	What is Cos2x?
ii.	What is Cos2x Formula in Trigonometry?
three.	Derivation of Cos2x Using Angle Add-on Formula
iv.	Cos2x In Terms of sin x
5.	Cos2x In Terms of cos x
6.	Cos2x In Terms of tan x
seven.	Cos^2x (Cos Square x)
8.	Cos^2x Formula
9.	How to Apply Cos2x Identity?
10.	FAQs on Cos2x

What is Cos2x?

Cos2x is an of import trigonometric function that is used to observe the value of the cosine part 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 issues. Cos2x is a double angle trigonometric function that determines the value of cos when the angle 10 is doubled.

What is Cos2x Formula in Trigonometry?

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

cos2x = cos^twoten - sin^twox
cos2x = 2cos²x - 1
cos2x = ane - 2sinⁱⁱten
cos2x = (1 - tanⁱⁱ10)/(i + tan²x)

Derivation of Cos2x Formula Using Bending Improver Formula

Nosotros know that the cos2x formula can be expressed in four dissimilar forms. Nosotros will utilise the angle improver formula for the cosine part to derive the cos2x identity. Annotation that the bending 2x can be written as 2x = ten + x. As well, 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 improver formula for cosine office, substitute a = b = ten into the formula for cos (a + b).

cos2x = cos (10 + x)

= cos ten cos x - sin x sin ten

= cos^twox - sinⁱⁱx

Hence, we accept cos2x = cos^twox - sin^twox

Cos2x In Terms of sin x

Now, that nosotros accept derived cos2x = cos^twox - sin²ten, nosotros will derive the formula for cos2x in terms of sine function only. We will use the trigonometry identity cos²ten + sin^twox = 1 to prove that cos2x = 1 - 2sin^two10. We take,

cos2x = cos²ten - sin²10

= (1 - sin²x) - sinⁱⁱten [Because cos²ten + sinⁱⁱx = 1 ⇒ cosⁱⁱ10 = 1 - sinⁱⁱ10]

= 1 - sin²x - sinⁱⁱx

= i - 2sin²x

Hence, we have cos2x = 1 - 2sinⁱⁱ10 in terms of sin x.

Cos2x In Terms of cos x

Just like nosotros derived cos2x = one - 2sinⁱⁱx, we will derive cos2x in terms of cos x, that is, cos2x = 2cos^twox - 1. We will use the trigonometry identities cos2x = cos²x - sinⁱⁱ10 and cos²10 + sin²x = i to prove that cos2x = 2cos^twox - 1. We have,

cos2x = cos²x - sin^twoten

= cos²x - (1 - cos²10) [Because cos^two10 + sinⁱⁱx = 1 ⇒ sin^twox = 1 - cos²x]

= cos^twox - 1 + cosⁱⁱ10

= 2cos²x - one

Hence , we have cos2x = 2cosⁱⁱ10 - 1 in terms of cosx

Cos2x In Terms of tan x

Now, that we have derived cos2x = cos^twox - sin^twox, we will derive cos2x in terms of tan x. We will employ a few trigonometric identities and trigonometric formulas such equally cos2x = cos²x - sin²x, cosⁱⁱx + sin^twox = ane, and tan x = sin x/ cos x. We have,

cos2x = cos²x - sin²x

= (cos²x - sin²x)/ane

= (cos²x - sin^two10)/( cos²x + sin²10) [Considering cosⁱⁱten + sin²10 = ane]

Separate the numerator and denominator of (cos²x - sin²x)/( cos²10 + sin²x) by cos²x.

(cos^two10 - sinⁱⁱten)/(cos^twox + sin²x) = (cos²x/cos²x - sin²10/cos^two10)/( cos²10/cos^twox + sin²ten/cos²x)

= (one - tan²ten)/(i + tanⁱⁱx) [Because tan x = sin x / cos x]

Hence, we have cos2x = (1 - tan²10)/(1 + tan²x) in terms of tan x

Cos^2x (Cos Square x)

Cos^2x is a trigonometric part that implies cos x whole squared. Cos square ten tin exist expressed in dissimilar forms in terms of different trigonometric functions such every bit cosine office, and the sine function. We will use unlike trigonometric formulas and identities to derive the formulas of cos^2x. In the next section, permit us get through the formulas of cos^2x and their proofs.

Cos^2x Formula

To get in at the formulas of cos^2x, we will use various trigonometric formulas. The showtime formula that we will use is sin^2x + cos^2x = one (Pythagorean identity). Using this formula, decrease sin^2x from both sides of the equation, we have sin^2x + cos^2x -sin^2x = ane -sin^2x which implies cos^2x = ane - 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 + ane)/ii. Therefore, the formulas of cos^2x are:

cos^2x = 1 - sin^2x ⇒ cos²x = one - sinⁱⁱx
cos^2x = cos2x + sin^2x ⇒ cos²ten = cos2x + sin^twox
cos^2x = (cos2x + 1)/2 ⇒ cos²ten = (cos2x + 1)/2

How to Employ Cos2x Identity?

Cos2x formula can exist used for solving different math problems. Let usa consider an case to understand the application of cos2x formula. We will determine the value of cos 120° using the cos2x identity. We know that cos2x = cos²10 - sin²x and sin 60° = √3/2, cos 60° = ane/2. Since 2x = 120°, x = sixty°. Therefore, we have

cos 120° = cos²60° - sin^two60°

= (1/2)² - (√three/ii)ⁱⁱ

= 1/4 - 3/four

= -one/two

Important Notes on Cos 2x

cos2x = cos²x - sin^two10
cos2x = 2cos²x - 1
cos2x = 1 - 2sinⁱⁱ10
cos2x = (1 - tan²x)/(ane + tan²ten)
The formula for cos^2x that is commonly used in integration problems is cos^2x = (cos2x + ane)/two.
The derivative of cos2x is -2 sin 2x and the integral of cos2x is (ane/2) sin 2x + C.

☛ Related Articles:

Trigonometric Ratios
Trigonometric Table
Sin2x Formula
Changed Trigonometric Ratios

Cos2x Examples

Instance ane: Prove the triple bending identity of cosine part using cos2x formula.

Solution: The triple angle identity of the cosine office is cos 3x = 4 cos³x - 3 cos x

To begin with, we will apply the angle addition formula of the cosine role.

cos 3x = cos (2x + 10) = cos2x cos x - sin 2x sin x

= (2cos²ten - 1) cos x - 2 sin ten cos ten sin ten [Because cos2x = 2cos²ten - one and sin2x = ii sin x cos x]

= 2 cos³x - cos x - 2 sin²ten cos x

= ii cosⁱⁱⁱx - cos x - 2 cos x (1 - cos²ten) [Considering cos²x + sin^twox = i ⇒ sin^twox = one - cosⁱⁱx]

= 2 cos^threeten - cos x - 2 cos x + two cos³x

= 4 cos³x - 3 cos x

Answer: Hence, we have proved cos 3x = iv cosⁱⁱⁱ10 - 3 cos x using the cos2x formula.
Example 2: Limited the cos2x formula in terms of cot x.

Solution: We know that cos2x = (one - tan²x)/(one + tan²10) and tan x = 1/cot ten

cos2x = (i - tan²10)/(1 + tan²x)

= (1 - i/cot^twox)/(1 + ane/cot²x)

= (cot²x - i)/(cotⁱⁱx + 1)

Answer: Hence, cos2x = (cotⁱⁱx - ane)/(cot²x + 1) in terms of cotangent function.
Example three: Decide 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 × two

= -2 sin 2x

To find the integral of cos2x, presume that 2x = u. And so 2 dx = du (or) dx = du/2. Substituting these values in the integral ∫ cos2x dx,

∫ cos2x dx = ∫ cos u (du/2)

= (ane/2) ∫ cos u du

Nosotros know that the integral of cos x is sin x + C. So,

(ane/ii) ∫ cos u du = (1/2) sin u + C

= (1/2) sin2x + C

Answer: Hence, integral of cos2x is (one/two) sin2x + C and its derivative of cos2x is -2 sin2x.

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FAQs on Cos2x

What is Cos2x Identity in Trigonometry?

Cos2x is one of the double angle trigonometric identities equally the angle in consideration is a multiple of 2, that is, the double of ten. It tin can be expressed in terms of different trigonometric functions such equally 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 tin can be expressed every bit:

cos2x = cos²10 - sin^twox
cos2x = 2cos²x - i
cos2x = 1 - 2sin²10

What is the Derivative of cos2x?

The derivative of cos2x is -two 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 tin can be easilty obtained using the formula ∫cos(ax + b) dx = (one/a) sin(ax + b) + C. Therefore, the integral of cos2x is given past ∫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 past cos2x = 1 - 2sin²x in terms of sin 10.

What is Cos2x In Terms of tan x?

We can express the cos2x formula in terms of tanx. The formula is given by cos2x = (one - 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 role, cos²x + sin²x = 1, tan ten = sin x/ cos x, etc.

How to Derive Cos Square x Formula?

We can derive the cos square ten formula using diverse 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. Nosotros can simplify these formulas and decide the value of cos square 10.

What is Cos^2x Formula?

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

cos^2x = ane - sin^2x ⇒ cos²x = 1 - sin²x
cos^2x = cos2x + sin^2x ⇒ cos²ten = cos2x + sin²x
cos^2x = (cos2x + 1)/2 ⇒ cos^twox = (cos2x + 1)/ii

What is the Formula of Cos2x in Terms of Cos?

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