## Cos 2x

The cos2x identity is an essential trigonometric formula used to find the value of the cosine function for double angles, also known as the double angle identity of the cosine function. This identity helps express the cosine of a compound angle 2x in various ways: in terms of sine and cosine functions, only the cosine function, only the sine function, and only the tangent function. Derived using different trigonometric identities, the cos2x formula can be represented in multiple forms. Additionally, we will explore the concept of cosĀ²x (cos square x) and its formula in detail.

## What is Cos2x?

Cos2x, also known as the double angle identity for cosine, is a trigonometric formula that expresses the cosine of a double angle (2x) using various trigonometric functions. It can be represented in multiple forms: cos 2x = cosĀ² x – sinĀ² x, cos 2x = 2 cosĀ² x – 1, cos 2x = 1 – 2 sinĀ² x, and cos 2x = (1 – tanĀ² x) / (1 + tanĀ² x). These identities are derived using the sum of angles formula and Pythagorean identities. The cos2x identity is essential for solving trigonometric equations, simplifying expressions, and analyzing periodic functions, making it a fundamental tool in mathematics and its applications in fields like physics, engineering, and computer science.

### Cos2x Identity

**In Terms of Cosine and Sine:**- Cos 2x = cosĀ² x – sinĀ² x

**In Terms of Cosine Only:**- Cos 2x = 2 cosĀ² x – 1

**In Terms of Sine Only:**- Cos 2x = 1 – 2 sinĀ² x

**In Terms of Tangent:**- Cos 2x = (1 – tanĀ² x) / (1 + tanĀ² x)

## Cos2x In Terms of sin x

- Start with the double-angle identity:

Cos2x = CosĀ²x – SinĀ²x - Use the trigonometric identity CosĀ²x + SinĀ²x = 1 to express CosĀ²x in terms of SinĀ²x:

CosĀ²x = 1 – SinĀ²x - Substitute this into the double-angle formula:

Cos2x = (1 – SinĀ²x) – SinĀ²x - Simplify the expression:

Cos2x = 1 – SinĀ²x – SinĀ²x - Combine like terms:

Cos2x = 1 – 2SinĀ²x

Hence, the formula for Cos2x in terms of Sin x is:

Cos2x = 1 – 2SinĀ²x

## Cos2x In Terms of cos x

- Start with the double-angle identity:

Cos2x = CosĀ²x – SinĀ²x - Use the trigonometric identity SinĀ²x = 1 – CosĀ²x to express SinĀ²x in terms of CosĀ²x:

SinĀ²x = 1 – CosĀ²x - Substitute this into the double-angle formula:

Cos2x = CosĀ²x – (1 – CosĀ²x) - Simplify the expression:

Cos2x = CosĀ²x – 1 + CosĀ²x - Combine like terms:

Cos2x = 2CosĀ²x – 1

Hence, the formula for Cos2x in terms of Cos x is:

Cos2x = 2CosĀ²x – 1

## Cos2x In Terms of tan x

- Start with the double-angle identity:

Cos2x = CosĀ²x – SinĀ²x - Use the trigonometric identities CosĀ²x = 1 / (1 + TanĀ²x) and SinĀ²x = TanĀ²x / (1 + TanĀ²x)Ā²:

Cos2x = (1 / (1 + TanĀ²x)) – (TanĀ²x / (1 + TanĀ²x)) - Express both terms with a common denominator:

Cos2x = (1 – TanĀ²x) / (1 + TanĀ²x)

Hence, the formula for Cos2x in terms of Tan x is:

Cos2x = (1 – TanĀ²x) / (1 + TanĀ²x)

## Properties of Cos2x

### Periodicity

Cos2x has a period of Ļ. This means that the function repeats its values every Ļ units. For example, Cos2(x + Ļ) is equal to Cos2x.

### Symmetry

Cos2x is an even function, meaning it is symmetric about the y-axis. This implies that Cos2(-x) is equal to Cos2x.

### Range

The range of Cos2x is between -1 and 1, inclusive. In other words, Cos2x will always produce values within this interval.

### Critical Points

Cos2x has critical points where its derivative is zero. These points occur at x = nĻ/2 for integers n. These points correspond to the local maxima and minima of the function.

### Maxima and Minima

Cos2x achieves its maximum value of 1 and its minimum value of -1 at specific points:

- Cos2x = 1 when x = nĻ for integers n.
- Cos2x = -1 when x = Ļ/2 + nĻ for integers n.

### Relation to Other Functions

Cos2x can be expressed in terms of other trigonometric functions:

- In terms of Cos x: Cos2x = 2CosĀ²x – 1
- In terms of Sin x: Cos2x = 1 – 2SinĀ²x
- In terms of Tan x: Cos2x = (1 – TanĀ²x) / (1 + TanĀ²x)

### Derivatives and Integrals

- The derivative of Cos2x is -2Sin2x.
- The integral of Cos2x is (1/2) Sin2x + C, where C is the constant of integration.

## Cos2x in relation to Other Trigonometric Functions

### In Terms of Cosine (Cos x)

The double-angle formula for cosine can be expressed using only the cosine function: Cos2x = 2CosĀ²x – 1

### In Terms of Sine (Sin x)

Cos2x can also be written in terms of the sine function: Cos2x = 1 – 2SinĀ²x

### In Terms of Tangent (Tan x)

To express Cos2x using the tangent function, we use the identity involving tangent: Cos2x = (1 – TanĀ²x) / (1 + TanĀ²x)

### In Terms of Secant (Sec x)

Though less common, Cos2x can be related to the secant function as well: Cos2x = (2 – SecĀ²x) / SecĀ²x

### In Terms of Cosecant (Csc x)

Cos2x can also be expressed in terms of the cosecant function: Cos2x = (2CscĀ²x – 1) / CscĀ²x

### In Terms of Cotangent (Cot x)

Finally, the double-angle formula for cosine can be written using the cotangent function: Cos2x = (CotĀ²x – 1) / (CotĀ²x + 1)

## Solved Problems

**Problem 1: Evaluating cosā”2x given cosā”x**

**Question:** Given cosā”x = 1/2, find cosā”2x

**Solution:** Using the double-angle formula for cosine: cosā”2x = 2cosā”Ā²xā1.

Substitute cosā”x=1/2:

cos2x = 2 (1/2)Ā² – 1 = 2 (1/4) -1 = 1/2-1 = -1/2

So, cos2x = -1/2

**Problem 2: Evaluating cosā”2x given sinā”x**

**Question:** Given sinā”x=**ā**3/2āā, find cosā”2x.

**Solution:** First, use the Pythagorean identity to find cosā”x: sinĀ²x+cosā”Ā²x = 1.

So, (**ā**3/2)Ā²+cosĀ²x=1.

This gives 3/4+cosā”Ā²x = 1, thus cosĀ²x =1/4, meaning cosā”x=Ā±1/2ā.

Using cosā”x = 1/2 or cosā”x=ā1/2ā:

cosā”2x = 2(1/2)Ā²ā1 = 1/2ā1 = ā1/2.

So, cosā”2x = ā1/2ā.

**Problem 3: Solving cosā”2x=0.5**

**Question:** Solve for cosā”2x = 0.5.

**Solution:** We know that cosā”2x=0.5.

This corresponds to: 2x=Ā±Ļ3+2kĻāforākāZ.

Thus: x=Ā±Ļ/6+kĻ.

So, the solutions are: x=Ļ/6+kĻ or x=āĻ/6+kĻ.

**Problem 4: Maximum and Minimum Values of cosā”2x**

**Question:** What are the maximum and minimum values of cosā”2x?

**Solution:** The cosine function oscillates between -1 and 1. Since cosā”2x is just a cosine function with a different argument, it also oscillates between -1 and 1.

Therefore:

Maximum value of cosā”2x is 1.

Minimum value of cosā”2x is -1.

**Problem 5: Expressing cosā”2x in terms of tanā”x**

**Question:** Express cosā”2x in terms of tanā”x

**Solution:** Using the identity cosā”2x=1ātanā”Ā²x/1+tanā”Ā²xā:

We start from the double-angle identity: cosā”2x = 2cosā”Ā²xā1.

Using cosā”Ā²x=1/1+tanĀ²xā:

cosā”2x = 2(1/1+tanā”Ā²x)ā1 = 2/(1+tanā”Ā²x)ā1 = 2ā(1+tanĀ²x)/(1+tanā”Ā²x) = (1ātanā”Ā²x)/(1+tanĀ²x).

So, cosā”2x=1ātanā”Ā²x/1+tanā”Ā²x.