Cos2x Formula: Cos2x Identity & Solved Examples

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Cos2x is a double-angle formula in Trigonometry that is used to find the value of the Cosine Function for double angles, where the angle is twice that of x. It is one of the double angle trigonometric identities as the angle is a multiple of 2, or, double of x.

The Cosine Function or cos defines the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
Cos2x Formula is used to determine the value of cos when the angle x is doubled.

The 4 different forms Cos2x Formula can be expressed as follows:

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

These variations of the Cos2x formula offer flexibility in solving trigonometric problems involving double angles.

Table of Content

What is Cos2x?
Cos2x Formula
Derivation of Cos2x Formula
What is Cos²x?
Things to Remember
Previous Years’ Questions
Sample Questions

Key Terms: Cos2x, Cos 2x Formula, Double Angle Formula, Cos2x Identity, Trigonometry, Integration, Trigonometric Functions

What is Cos2x?

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Cos2x is a trigonometric function used to find the value of the cosine function for the compound angle 2x.

Cos2x is a double-angle trigonometric function used to find the value of cos when the angle x is doubled.
It can be expressed in terms of different trigonometric functions such as Sine Function, and Tangent Function.
Cos2x Formula is used to simplify complex trigonometric expressions and solve integration problems.
Cos2x Identity can be derived with the help of different trigonometric identities.

Double-angle formulas simplify expressions for solving complex problems involving angles. They provide a systematic way to express trigonometric functions when dealing with angles that are twice the size of another angle.

Trigonometric Functions Detailed Video Explanation

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Cosine Rule	Sin 180 Degrees	Law of Sines
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Cos2x Formula

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

Cos2x Formula is written in different forms as follows:

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

Solved Example

Example: Using Cos2x Formula, Prove the Triple Angle Identity of the Cosine Function.

Solution: Triple Angle Identity of the Cosine Function is cos 3x = 4 cos³x - 3 cos x.

Using the Angle Addition Formula of the Cosine Function, we get

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

= (2cos²x - 1) cos x - 2 sin x cos x sin x [As 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

Thus, the Triple Angle Identity of the Cosine Function, i.e. cos 3x = 4 cos³x - 3 cos x is proved using the Cos2x Formula.

Derivations of Cos2x Formula

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The derivation of the cos2x formula involves applying the angle addition formula for the cosine function, which is cos(a+b) = cosa cosb − sina sinb. Angle 2x can be written as 2x = x + x.

By substituting a = b = x into this identity, the cos2x formula can be systematically derived. Here's a step-by-step explanation to enhance comprehension:

Start with the angle addition formula: cos(a+b)=cosacosb−sinasinb.
Substitute a = b = x into the formula to obtain cos²x=cos²x−sin²x.
Simplify further using trigonometric identities to reach the standard forms of the cos2x formula.

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

Thus, the general Cos2x Formula cos2x = cos²x - sin²x, is derived.

Cosine Function

Cosine Function

Cos2x In Terms of Sinx

Cos2x Formula in terms of the Sine function is cos2x = 1 - 2sin²x. It can be proved using the trigonometry identity cos²x + sin²x = 1.

We have,

cos2x = cos²x - sin²x

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

= 1 - sin²x - sin²x

= 1 - 2sin²x

Thus, Cos2x Formula in terms of Sine is cos2x = 1 - 2sin²x.

Cos2x In Terms of Cosx

Cos2x Formula in terms of Cosx is cos2x = 2cos²x - 1. It can be derived using the trigonometric identities, cos2x = cos²x - sin²x and cos²x + sin²x = 1.

We have

cos2x = cos²x - sin²x

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

= cos²x - 1 + cos²x

= 2cos²x - 1

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

Cos2x In Terms of Tanx

Cos2x Formula in terms of Tangent Function is cos2x = (1 - tan²x)/(1 + tan²x). It can be derived using various trigonometric identities and trigonometric formulas such as

cos2x = cos²x - sin²x
cos²x + sin²x = 1
tan x = sin x/ cos x

We have

cos2x = cos²x - sin²x

= (cos²x - sin²x)/1

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

Now, 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]

Thus, Cos2x Formula in terms of Tan is cos2x = (1 - tan²x)/(1 + tan²x).

Cos2x Formula Solved Examples

Here are a few solved examples on the Cos2x Formula for a better understanding of the formula:

Example 1: Prove Cos3x = 4Cos3x– 3Cosx.

Solution: Taking LHS,

Cos 3x can be written as Cos3x = Cos (2x+x) ….. (1)

Using Trigonometric Identity cos(a+b) = cos(a)cos(b) - sin(a)sin(b) in equation (1), we get

= Cos2x Cos x –Sin 2x Sin x

= (2Cos2x–1) Cos x –2Sin x Cosx Sin x

= (2Cos2x –1) Cos x –2Cos x (1– Cos2x)

= 2Cos3x –Cos x –2Cos x +2Cos3x

= 4Cos3x –3Cos

Thus, LHS = RHS

Example 2: Express Cos2x Formula in terms of Cot x.

Solution: It is known 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)

Thus, Cos2x Formula in terms of Cot x is cos2x = (cot²x - 1)/(cot²x + 1).

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What is Cos²x?

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Cos²x or Cos Square x is a trigonometric function that denotes that cos x is whole squared. Cos²x can be expressed in different forms in terms of different trigonometric functions such as cosine and sine functions. Cos²x Formula can be derived using different trigonometric formulas and identities.

Cos²x Formula

To derive Cos²x Formula, the identity sin²x + cos²x = 1 (Pythagorean Identity) is used.

Subtract sin²x from both sides of the equation, to get sin²x + cos²x -sin²x = 1 -sin²x which implies cos²x = 1 - sin²x.
cos2x = cos²x - sin²x and cos2x = 2cos²x - 1 are the two Cos2x Formulas that include Cos²x.
Using these formulas, cos²x = cos2x + sin²x and cos²x = (cos2x + 1)/2 are obtained.

Thus, the formulas of Cos²x are as follows:

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

Things to Remember

The Cos 2x formula is derived from fundamental trigonometric identities, particularly the Pythagorean identity, which states that sin²x + cos²x = 1.
Cos2x is a double-angle formula of the cosine function in Trigonometry.
One form of the Cos 2x formula is cos²x - sin²x, which can be rearranged as cos 2x = cos²x - sin²x.
- In terms of Sine: cos 2x = 1 - 2sin²x
- In terms of Cosine: cos 2x = 2cos²x - 1
- In terms of Tangent: cos 2x = (1 - tan²x)/(1 + tan²x).
Cos 2x is useful in double-angle trigonometry to calculate cosine values for angles twice the size of x.
Cos2x formula helps to understand geometric relationships in right-angled triangles, defining the ratio of the adjacent side to the hypotenuse.

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Previous Years’ Questions

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Sample Questions

Ques. Prove that 2 Cos 4x + 1 / 2 Cos x + 1 = (2 Cos x – 1) (2 Cos 2x – 1). (3 Marks)

Ans. The given trigonometric equation is 2 Cos 4x + 1 / 2 Cos x + 1 = (2 Cos x – 1) (2 Cos 2x – 1).

LHS is given as:

= 2 Cos 4x + 1 / 2 Cos x + 1

= 2[2 Cos² (2x) – 1] + 1 / 2 Cos x +1

= 4 Cos² 2x – 2 +1 / 2 Cos x +1

= 4 Cos² 2x – 1 / 2 Cos x +1

= [2 Cos 2x]² - 12 / 2 Cos x +1

= [2 Cos²x +1] [ 2 Cos 2x -1] / 2 Cos x +1

= [2 (2Cos²x – 1) + 1] [ 2 Cos 2x -1] / 2 Cos x +1

= [4 Cos²x – 2 + 1] [ 2 Cos 2x -1] / 2 Cos x +1

= [(2 Cos x)² – 1²] [ 2 Cos 2x -1] / 2 Cos x +1

= [ 2 Cos x +1] [ 2 Cos x -1] [ 2 Cos 2x -1] / 2 Cos x +1

= (2 Cos x – 1) (2 Cos 2x – 1)

Thus, LHS = RHS.

Hence Proved.

Ques. Prove that Cos²A + Cos²(A+1200) + Cos² (A+1200) = 3/2. (3 Marks)

Ans. The given trigonometric equation is Cos²A + Cos²(A+1200) + Cos² (A+1200) = 3/2.

LHS is given as:

= Cos²A + Cos² (A+120⁰) + Cos² (A+120⁰)

= 3/2 + 1/2 [ Cos 2A + Cos (2A+240⁰) + Cos (2A+240⁰)]

= 3/2 + 1/2 [ Cos 2A + 2 Cos 2A Cos 240⁰]

= 3/2 + 1/2 [ Cos 2A + 2 Cos 2A Cos (180⁰ + 60⁰)]

= 3/2 + 1/2 [ Cos 2A - 2 Cos 2A Cos 60⁰]

= 3/2 + 1/2 [ Cos 2A - Cos 2A]

= 3/2 + 1/2 [0]

= 3/2

Thus, LHS = RHS.

Hence Proved.

Ques. Determine the Derivative and Integral of Cos2x. (3 Marks)

Ans. Using 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

Thus, the integral of cos2x is (1/2) sin2x + C and its derivative of cos2x is -2 sin2x.

Ques. What is Cos2x Formula? (3 Marks)

Ans. Cos2x Formula is a double-angle formula in trigonometry that is used to find the value of the cosine trigonometric function for double angles. It can be expressed in terms of different trigonometric functions such as sine, cosine, and tangent. It can be expressed in four different forms as follows:

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

Ques. If cos x = 3/5, find the value of Cos2x using the Cos2x Formula. (2 Marks)

Ans. We have, cos x = 3/5.

Clearly, sin x = 4/5.

Using the Cos2x Formula, we get,

cos 2x = cos2 x – sin2 x

= (3/5)2 – (4/5)2

= 9/25 – 16/25

= -7/25

Ques. If sin x = 3/5, find the value of Cos2x using the Cos2x Formula. (2 Marks)

Ans. We have, sin x = 3/5.

Cos x = 4/5.

Using the Cos2x Formula, we get,

cos ²x = cos² x – sin² x

= (4/5)2 – (3/5)2

= 16/25 – 9/25

= 7/25

Ques. If sec x = 17/8, find the value of Cos2x using the Cos2x Formula. (2 Marks)

Ans. We have Sec x = 17/8.

Cos x = 8/17 and sin x = 15/17.

Using the Cos2x Formula, we get,

cos 2x = cos² x – sin² x

= (8/17)² – (15/17)²

= 64/289 – 225/289

= -161/225

Ques. If cos² x = 5/8, find the value of Cos2x using the Cos2x Formula. (2 Marks)

Ans. We have,cos² x = 5/8

Using the Cos2x Formula, we get,

cos 2x = 2 cos² x – 1

= 2 (5/8) – 1

= 5/4 – 1

= 1/4

Ques. Find the value of Cos 120° using the Cos2x Formula. (2 Marks)

Ans. It is known that cos2x = cos²x - sin²x and sin 60° = √3/2, cos 60° = 1/2. As, 2x = 120°, and x = 60°.

cos 120° = cos²60° - sin²60°

= (1/2)² - (√3/2)²

= 1/4 - 3/4

= -1/2

Thus, the value of cos 120° is -½.

Ques. If Sin x = 12/13, Find Cos 2x. (2 Marks)

Ans. We know that,

Cos2x = 1 – 2Sin2x

= 1 – 2 (12/13)²

= 1 – 2 (144/169)

= 1 –288/ 169

= 169 – 288/169

= -119/169

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Cos2x Formula: Cos2x Identity & Solved Examples

What is Cos2x?