Tan2x Formula: Tan2x Identity, Proof & Solved Examples

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Tan2x Formula is a commonly used double-angle trigonometric formula that is used to find the Tangent of the angle with a double value. It is one of the three important double-angle formulas including the Sin2x Formula and Cos2x Formula that have double angles in their trigonometric functions.

Tangent Function is the ratio of the opposite side and the adjacent side of a right-angled triangle.
It is abbreviated as ‘Tan’ and is given as Tan x = Opposite Side/Adjacent Side = Perpendicular/Base.
Tan2x Formula is expressed in terms of different trigonometric functions such as tan x, cos x, and sin x.
It can also be expressed as the ratio of sin2x and cos2x.

Tan2x Formula is given as:

Tan2x = 2Tanx / (1 − Tan²x)

It can also be expressed as:

Tan2x = Sin2x/Cos2x

Table of Content

What is Tan2x?
Tan2x Formula
Tan2x Formula Proof
Tan2x Graph
Tan2x Formula Solved Examples
What is Tan²x?
Things to Remember
Previous Years’ Questions
Sample Questions

Key Terms: Tan2x Formula, Tangent Function, Sine Function, Cosine Function, Trigonometric Function, Double Angle Formula

What is Tan2x?

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Tan2x is an important double-angled trigonometric function that is used to solve a wide range of trigonometric and integration problems.

Tan2x is an important double-angle formula, that is used to find the value of tan when the angle is doubled.
Tan2x Formula can be expressed in terms of Tan x and also as a ratio of sin2x and cos2x.
Since the reciprocal of tan x is cot x, thus, tan2x can be written as the reciprocal of cot 2x, i.e. Tan2x = 1/Cot2x.
The value of Tan2x repeats after every 2π radians, i.e. Tan2x = tan (2x + 2π).
The graph of Tan2x is noticeably thinner than the Tan x graph.

Read More:

Relevant Concepts
Value of Cos 180 Degrees	Sin 180 Degrees	Degrees to Radians
Cosine Rule	Domain and Range of Trigonometric Functions	Law of Sines

Tan2x Formula

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Tan2x Formula is a double-angle formula used for Tangent Function in Trigonometry. It can be expressed in terms of tangent function only and as a combination of the sine function and cosine function. Thus, Tan2x Formula is expressed in two ways.

Tan2x Formula in terms of Tan Function is given as:

Tan2x = 2Tanx / (1 − Tan²x)

Tan2x Formula in terms of Sin2x and Cos2x is given as:

Tan2x = Sin2x/Cos2x

Tan2x Formula Proof

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Tan2x Formula can be derived using two different methods in Trigonometry. First, Tan2x Formula can be derived using the angle addition formula for the tangent function.

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

We have,

Tan2x = Tan (x + x)

= (tan x + tan x)/(1 - tan x tan x)

= 2 tanx/(1 - tan²x)

Thus, the Tan2x Formula has been derived using the angle sum formula of the tangent function.

Tan2x Identity Proof Using Sin and Cos

Tan2x Formula can also be derived by expressing tan as a ratio of sin and cos. We know that,

Tan x = sin x/ cos x
Sin 2x = 2 sin x cos x
Cos 2x = cos²x - sin²x

Using the above-mentioned formulas, we get

Tan2x = Sin 2x/Cos 2x

= 2 sin x cos x/(cos²x - sin²x)

Now, divide the numerator and denominator 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)

Thus, the Tan2x Formula has been derived by expressing it as a ratio of sin 2x and cos 2x.

Trigonometric Functions Detailed Video Explanation

Tan2x Graph

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Tan2x Graph is similar to the graph of Tan x in appearance.

The period of Tan x is π and the period of tan2x is π/2.
The value of tan2x repeats after every π/2 radians.
The value of tanx is equal to zero whenever x is an integral multiple of π.
Tan2x is equal to zero whenever 2x = nπ, where n is an integer.

Tan2x Graph

Tan2x Formula Solved Examples

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Here are a few solved examples on Tan2x Formula for a better understanding of the formula:

Example: Find the value of Tan2x if tan x = 12/5.

Solution: It is given that tan x = 12/5.

Using the Tan2x Formula,

Tan2x = 2Tan x / (1−Tan²x)

= (2 (12/5))/(1 – (12/5)²)

= (24/5)/(1 – 144/25)

= (24/5)/(-119/25)

= -120/119

Thus, the value of Tan2x when tan x = 12/5 is -120/119.

Example 2: Find the value of Tan2x if sin x = 12/13 and cos x = 5/13.

Solution: Tan2x Formula in terms of Sin2x and Cos2x is given as Tan2x = Sin 2x/Cos 2x.

On expanding the Sin 2x and Cos 2x Formulas, we get

= 2 sin x cos x /(cos²x - sin²x)

= [2 × (12/13) × (5/13)] / [(25/169) - (144/169)]

= (120/169) / (-119/169)

= -120/119

Thus, Tan2x is equal to -120/119 when sin x = 12/13 and cos x = 5/13.

What is Tan²x?

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Tan²x or Tan Square x is the square of the trigonometric function Tan x. It can be derived using the trigonometric identities and formulas which consist of tan²x. Tan x is expressed as the ratio of sin x and cos x, thus, tan²x can also be expressed as the ratio of sin square x and cos square x. Tan²x Formula is used to solve complex integration and differentiation problems and simplify trigonometric expressions.

Tan²x Formula

Tan x is expressed as the ratio of the sine function and cosine function. Thus, Tan square x can be expressed as the ratio of sin square x and cos square x. Thus, Tan²x Formulas are listed as follows:

Tan²x = sec²x - 1 ⇒ tan²x = sec²x - 1
Tan²x = sin²x / cos²x ⇒ tan²x = sin²x/cos²x
Tan²x = 1/cot²x ⇒ tan²x = 1/cot²x

Check More:

Related Topics
Cosine Formula	Side Angle Side Formula	Cotangent Formula
Degree and Radian Measure Formula	Sin Squared X Formula	2 Cos A Cos B Formula
Tangent 3 Theta formula	Cos Square Theta formula	Cot Tan Formula

Things to Remember

Tangent Function is a trigonometric ratio which is the ratio of the perpendicular and base of a right-angled triangle.
Tan2x Formula is a double-angle formula that is used to find the tangent of the angle whose value is doubled.
It is one of the important double-angle formulas in Trigonometry along with Sin2x and Cos2x.
Tan2x Formula in terms of Tan Function is Tan2x = 2Tan x / (1−Tan²x).
Tan2x Formula in terms of Sin2x and Cos2x is Tan2x = Sin 2x/Cos 2x.
It is used to solve a wide range of trigonometric and integration problems.

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Previous Years’ Questions

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Sample Questions

Ques. Determine the value of Tan2x if tan x = 3/4. (3 Marks)

Ans. It is given that tan x = 3/4.

Using the Tan2x Formula,

Tan2x = 2Tan x / (1−Tan²x)

Now, Tan²x = (3/4)² = 9/16

Substitute the value in the Tan2x Formula,

Tan2x = 2(3/4)/(1 - 9/16)

= (3/2)/(7/16)

= 24/7

Thus, Tan2x is equal to 24/7 when tan x = 3/4.

Ques. Calculate the tangent angle of a right triangle whose adjacent side and opposite sides are 10 cm and 8 cm respectively. (3 Marks)

Ans. Given that.

Adjacent Side = 10 cm
Opposite Side = 8 cm

Using Tangent Formula, we get

Tan θ = Opposite Side/Adjacent Side

Tan θ = 8/10

Tan θ = 0.8

Thus, the value of the tangent angle of a right triangle whose adjacent side and opposite sides are 10 cm and 8 cm is 0.8.

Ques. What is Tan Square X? (1 Mark)

Ans. Tan Square X is defined as the square of the trigonometric function Tan x and is mathematically expressed as tan²x. Tan²x Formula is used to solve complex integration and differentiation problems and simplify trigonometric expressions.

Ques. Determine the value of Tan2x if tan x = 4/3. (3 Marks)

Ans. It is given that tan x = 4/3.

Using the Tan2x Formula,

Tan2x = 2Tan x / (1−Tan²x)

Now, Tan²x = (4/3)² = 16/9

Substitute the value in the Tan2x Formula,

Tan 2x = 2tan x / (1 −tan²x)

= 2(4/3)/(1 - 16/9)

= (8/3)/(-7/9)

= -24/7

Thus, the value of Tan2x is -24/7 if tan x = 4/3.

Ques. Find the value of Tan2x if sin x = 9/13 and cos x = 4/13. (3 Marks)

Ans. Tan2x Formula in terms of Sin2x and Cos2x is Tan2x = Sin 2x/Cos 2x.

On expanding the Sin 2x and Cos 2x Formulas,

= 2 sin x cos x /(cos²x - sin²x)

= [2 × (9/13) × (4/13)] / [(16/169) - (81/169)]

= (72/169) / (-65/169)

= -72/65

Thus, the value of Tan2x is -72/65 if sin x = 9/13 and cos x = 4/13.

Ques. State the difference Between Tan2x and Tan²x. (2 Marks)

Ans. Tan2x is defined as a double-angle trigonometric formula that gives the value of the tangent function for the compound angle 2x. Whereas, Tan²x is the whole square of the trigonometric function Tan x. The value of Tan 2x can be both positive as well as negative whereas the value of Tan²x is always non-negative as the square of a number can never be never negative.

Ques. Calculate the tangent angle of a right triangle whose adjacent sides and opposite sides are 15 cm and 6 cm respectively. (3 Marks)

Ans. Given that,

Adjacent Side = 15 cm
Opposite Side = 6 cm

Using Tangent Formula, we get

Tan θ = Opposite Side/Adjacent Side

Tan θ = 6/15

Tan θ = 0.4

Thus, the tangent angle of a right triangle whose adjacent sides and opposite sides are 15 cm and 6 cm is 0.4.

Ques. Calculate the tangent angle of a right triangle whose adjacent side and opposite sides are 12 cm and 11 cm respectively. (3 Marks)

Ans. Given that,

Adjacent Side = 12 cm

Opposite Side = 11 cm

Using Tangent Formula, we get

Tan θ = Opposite Side/Adjacent Side

Tan θ = 11/12

Tan θ = 0.9167

Thus, the tangent angle of a right triangle whose adjacent side and opposite sides are 12 cm and 11 cm is 0.9167.

Ques. Determine the value of Tan2x if tan x = 2/3. (3 Marks)

Ans. It is given that tan x = 2/3.

Using the Tan2x Formula,

Tan2x = 2Tan x / (1−Tan²x)

Now, Tan²x = (2/3)² = 4/9

Substitute the obtained value in the Tan2x Formula,

Tan 2x = 2tan x / (1 −tan²x)

= 2(2/3)/(1 - 4/9)

= (4/3)/(5/9)

= 36/15

Thus, the value of Tan2x is 36/15 if tan x = 2/3.

Ques. In how many ways Tan2x Formula be expressed? (2 Marks)

Ans. Tan2x Formula can be expressed in the following ways:

Tan2x = 2tan x / (1 − tan²x)
Tan2x = sin 2x/cos 2x
Tan2x = [2 cos x/(2 cos²x - 1)]√(1 - cos²x)
Tan2x = [2 sin x/(1 - 2 sin²x)]√(1 - sin²x)

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Tan2x Formula: Tan2x Identity, Proof & Solved Examples

What is Tan2x?

Tan2x Formula

Tan2x Formula Proof

Tan2x Identity Proof Using Sin and Cos

Tan2x Graph

Tan2x Formula Solved Examples

What is Tan²x?

Tan²x Formula

Things to Remember

Previous Years’ Questions

Sample Questions

CBSE CLASS XII Related Questions

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

3.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

4.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

5.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

6.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

Similar Mathematics Concepts

Comments

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Tan2x Formula: Tan2x Identity, Proof & Solved Examples

What is Tan2x?

Tan2x Formula

Tan2x Formula Proof

Tan2x Identity Proof Using Sin and Cos

Tan2x Graph

Tan2x Formula Solved Examples

What is Tan2x?

Tan2x Formula

Things to Remember

Previous Years’ Questions

Sample Questions

CBSE CLASS XII Related Questions

1. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

3.Find:Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

4.Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

5.At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.On the basis of the above information, answer the following questions :

6.Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

What is Tan²x?

Tan²x Formula

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

3.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

4.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

5.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

6.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).