# Tan2x Formula

## Tan2x Formula

Trigonometry is a mathematical area that uses trigonometric ratios to calculate the angles and unknown sides of a triangle. It facilitates the estimate of unknown dimensions of a right-angled triangle by employing equations and identities based on this relationship. Trigonometry is defined by the terms ‘Trigonon’ and ‘Metron,’ which symbolise a triangle and a measurement, respectively. This field of mathematics is investigated utilising trigonometric ratios such as sine, cosine, tangent, cotangent, secant, and cosecant. It is the investigation of the connection between the sides and angles of a right-angled triangle.

A trigonometric ratio is the ratio of the lengths of any two sides of a right triangle. These ratios link the sides of a right triangle to their angles in trigonometry. The tangent ratio is determined by dividing the length of an angle’s opposing side by the length of its neighbouring side. It is abbreviated as tan.

What is Tan2x in Trigonometry?

Tan2x is a trigonometric function with a formula that can be used to solve different trigonometric issues. The Tan2x Formula is a useful double-angle formula or a trigonometry formula in which the angle is doubled. It can be stated in terms of tan x as well as a sin2x/cos2x ratio. Because the reciprocal of tan x is cot x, students can write tan2x as the reciprocal of cot 2x, i.e. tan2x = 1/cot2x.

## Tan2x Formula

The Tan2x Formula is a double-angle identity in trigonometry. The tangent function can alternatively be written as tan2x = sin 2x/cos 2x since it is a ratio of the sine and cosine functions. The Tan2x Formula is a crucial trigonometric identity that is used to solve a variety of trigonometric and integration issues. The value of tan2x repeats every 2π radians, tan2x = tan (2x + 2π). The Tan2x Formula has a considerably smaller graph than tan x. It is a trigonometric function that returns the tan function value of a double angle.

The Tan2x Formula can be expressed in two ways. It can be represented solely in terms of the tangent function or as a mix of the sine and cosine functions.

The Tan2x Formula identity is given as:

The Tan2x Formula  = 2tan x / (1−tan2x)

The Tan2x Formula = sin 2x/cos 2x

### Tan2x Formula Proof

The Tan2x formula can be obtained in two ways. First, students will obtain the tan2x identity using the angle addition formula for the tangent function. It is worth noting that the double angle 2x can be written as 2x = x + x. To prove the formula for tan2x, students will utilise the trigonometric formula:

tan (a + b) = (tan a + tan b)/(1 – tan a tan b)

they have

The Tan2x Formula = tan (x + x)

= (tan x + tan x)/(1 – tan x tan x)

= 2 tan x/(1 – tan2x)

As a result, they used the angle sum formula of the tangent function to obtain the Tan2x Formula.

### Tan2x Identity Proof Using Sin and Cos

A trigonometric ratio is the length ratio of any two sides of a right triangle. These ratios connect the ratio of sides of a right triangle to trigonometric angles. The tangent ratio is determined by dividing the length of an angle’s opposing side by the length of its neighbouring side. Tan is the acronym for it.

• tan x = sin x/ cos x
• sin 2x = 2 sin x cos x
• cos 2x = cos2x – sin2x

Using the above formulas, we have

tan2x = sin 2x/cos 2x

= 2 sin x cos x/(cos2x – sin2x)

Divide the denominator and numerator of 2 sin x cos x/(1 – 2 sin2x) by cos2x

tan2x = [2 sin x cos x/cos2x]/[(cos2x – sin2x)/cos2x]

= (2 sin x/cos x)/(1 – sin2x/cos2x)

= 2 tan x/(1 – tan2x)

### Tan2x Graph

The graph of tan2x resembles the graph of tan x. Students already know what the period of tan x is.π Because π/|b| gives the period of tan bx, the period of tan2x is π/2. The graph of tan2x is shown below, and as can be seen, the value of tan2x repeats every π/2 radian. Also, because tanx is equal to zero anytime x is an integral multiple of π, tan2x is equal to zero whenever 2x = nπ, where n is an integer, implying that the graph below has x-intercepts at x = nπ/2.

### Tan^2x (Tan Square x)

Tan^2x is the trigonometric function tanx squared. Using trigonometric identities and formulae including tan^2x, one can get the tan square x formulas. Because tan x can be represented as a ratio of sinx and cosx, they can define tan2x as a ratio of sin square x and cos square x. The tan^2x formula is used to handle difficult integration and differentiation issues as well as to simplify trigonometric formulas. The formula for tan square x will be deduced and discussed in the next section.

### Tan^2x Formula

Students now have a trigonometric identity 1 + tan^2x = sec^2x, implying tan^2x = sec^2x – 1. Because tan x can be defined as the ratio of the sine function and the cosine function, they can write tans square x as the ratio of sin square x and cos square x, yielding tan^2x = sin^2x / cos^2x. Also, because tan x is the reciprocal of cot x, they can write tan^2x = 1/cot^2x. As a result, the following tan^2x formulas are listed:

tan^2x = sec^2x – 1 ⇒ tan2x = sec2x – 1

tan^2x = sin^2x / cos^2x ⇒ tan2x = sin2x/cos2x

tan^2x = 1/cot^2x ⇒ tan2x = 1/cot2x

### Tan2x in Terms of Cos

In terms of cos, one can get the Tan2x Formula. To express tan2x in terms of cos x, they shall utilise the trigonometric formulae shown below.

tan x = sin x/ cos x

sin 2x = 2 sin x cos x

cos 2x = 2 cos2x – 1

sin x = √(1 – cos2x)

Student will have after using the above formulas,

The Tan2x Formula = sin 2x/ cos 2x

= 2 sin x cos x/(2 cos2x – 1)

= [2 √(1 – cos2x) cos x/(2 cos2x – 1)]

Similarly, they can write tan2x in terms of sin using the trigonometric identities.

The Tan2x Formula = [2 sin x/(1 – 2 sin2x)]√(1 – sin2x)

### Solved Examples

If tan x = 4/5, calculate the value of tan 2x.

Solution: We have, tan x = 4/5.

Using the formula we get,

tan 2x = 2 tan x/(1 – tan2 x)

= (2 (4/5))/(1 – (4/5)2)

= (8/5)/(1 – 16/25)

= (8/5)/(9/25)

= 72/125

## FAQs (Frequently Asked Questions)

### 1. What is the Tan2x Formula in trigonometry?

The Tan2x Formula is a fairly popular double-angle trigonometric formula that can be stated in terms of several trigonometric functions such as tan x, cos x, and sin x. Tan2x is generally defined as

The Tan2x Formula  = 2tan x / (1 tan2x).

### 2. What Is the Distinction Between Tan^^2x and Tan2x?

The Tan2x Formula is a trigonometric double-angle formula that calculates the tangent function for the compound angle 2x. Tan^2x, on the other hand, is the whole square of the trigonometric function tanx. The Tan2x Formula can have both positive and negative values, whereas tan2^x is always non-negative, since the square of an integer is never negative.

### 3. What is a Tan Function?

Normally, the Tangent Angle Formula is used to compute the angle of a right-angle triangle. The tangent of an angle in any given right triangle is the length of the opposing side divided by the length of the neigh bouring side. The tangent function is one of the three most popular trigonometric functions, along with sine and cosine. The tangent of an angle in any given right angle is the length of the opposing side divided by the length of the neigh bouring side.

Students can just write it as ‘tan’ in a formula.

The tan formula is calculated by dividing opposed sides by adjacent sides.

Trigonometric formulae such as Sin 2x, Cos 2x, and Tan 2x are known as double-angle formulas because their trigonometric functions include double angles.

There are two ways to represent the tan 2x expression. This can only be written as a tangent function and as a sine and cosine function combination. The formula for the tan 2x identity is as follows:

Tan 2x equals sin 2x/cos 2x