Trigonometric identities

     

Tan2x is an important trigonometric function. Tan2x formula is one of the very commonly used double angle trigonometric formulas and can be expressed in terms of different trigonometric functions such as rã x, cos x, and sin x. As we know that chảy x is the ratio of sine and cosine function, therefore the tan2x identity can also be expressed as the ratio of sin 2x and cos 2x.

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In this article, we will learn the tan2x & tan^2x formula, its proof, and express it in terms of different trigonometric functions. We will also explore the graph of tan2x and its period along with the concept of chảy square x & solve examples for a better understanding.

1.What is Tan2x in Trigonometry?
2.Tan2x Formula
3.Tan2x Formula Proof
4.Tan2x Graph
5.Tan^2x (Tan Square x)
6.Tan^2x Formula
7.Tan2x Formula in Terms of Cos
8.FAQs on Tan2x Formula

What is Tan2x in Trigonometry?


Tan2x is a trigonometric function & has a formula that is used khổng lồ solve various problems in trigonometry. Tan2x is an important double angle formula, that is, a trigonometry formula where the angle is doubled. It can be expressed in terms of chảy x and also as a ratio of sin2x and cos2x. Since the reciprocal of tung x is cot x, therefore we can write tan2x as the reciprocal of cot 2x, that is, tan2x = 1/cot2x. Let us see the tan2x formula:


Tan2x Formula


We can express the tan2x formula in two different forms. It can be expressed in terms of tangent function only và as a combination of the sine function & cosine function. The formula for tan2x identity is given as:

tan2x = 2tan x / (1−tan2x)tan2x = sin 2x/cos 2x

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Tan2x Formula Proof


Tan2x formula can be derived using two different methods. First, we will use the angle addition formula for the tangent function lớn derive the tan2x identity. Chú ý that we can write the double angle 2x as 2x = x + x. We will use the following trigonometric formula khổng lồ prove the formula for tan2x:

We have

tan2x = rã (x + x)

= (tan x + tung x)/(1 - rã x rã x)

= 2 tan x/(1 - tan2x)

Hence, we have derived the tan2x formula using the angle sum formula of the tangent function.

Tan2x Identity Proof Using Sin & Cos

Now, we will derive the tan2x formula by expressing tung as a ratio of sin and cos. We will use the following trigonometric formulas:

tan x = sin x/ cos xsin 2x = 2 sin x cos xcos 2x = cos2x - sin2x

Using the above formulas, we have

tan2x = sin 2x/cos 2x

= 2 sin x cos x/(cos2x - sin2x)

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)

Hence we have derived the tan2x formula by expressing it as a ratio of sin 2x and cos 2x.

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Tan2x Graph


The graph of tan2x looks similar lớn the graph of tan x. We know that the period of tung x is π. Since the period of tan bx is given by π/|b|, the period of tan2x is π/2. Given below is the graph of tan2x và as we can observe from the graph, the value of tan2x repeats after every π/2 radians. Also, the value of tanx is equal to zero whenever x is an integral multiple of π, therefore tan2x is equal to zero whenever 2x = nπ, where n is an integer which implies the below graph has x-intercepts at x = nπ/2.

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Tan^2x (Tan Square x)


Tan^2x is the square of the trigonometric function tanx. We can derive the tan square x formulas using the trigonometric identities and formulas which consist of tan^2x. As we know that tung x can be expressed as the ratio of sinx & cosx, therefore we can express tan^2x can as the ratio of sin square x và cos square x. We use the tan^2x formula to lớn solve complex integration & differentiation problems and simplify trigonometric expressions. In the next section, we will derive and discuss the formula of rã square x.


Tan^2x Formula


Now, we have a trigonometric identity 1 + tan^2x = sec^2x which implies tan^2x = sec^2x - 1. Since tung x can be expressed as the ratio of sine function and cosine function, therefore we can write tans square x as the ratio of sin square x và cos square x, therefore we have tan^2x = sin^2x / cos^2x. Also, we know that tung x is the reciprocal of cot x, therefore we can write tan^2x = 1/cot^2x. Hence, the danh sách of tan^2x formula is:

tan^2x = sec^2x - 1 ⇒ tan2x = sec2x - 1tan^2x = sin^2x / cos^2x ⇒ tan2x = sin2x/cos2xtan^2x = 1/cot^2x ⇒ tan2x = 1/cot2x

Tan2x in Terms of Cos


We can derive the tan2x formula in terms of cos. We will use the following trigonometric formulas to lớn express tan2x in terms of cos x.

tan x = sin x/ cos xsin 2x = 2 sin x cos xcos 2x = 2 cos2x - 1sin x = √(1 - cos2x)

Using the above formulas, we have

tan2x = sin 2x/ cos 2x

= 2 sin x cos x/(2 cos2x - 1)

= <2 √(1 - cos2x) cos x/(2 cos2x - 1)>

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

tan2x = <2 sin x/(1 - 2 sin2x)>√(1 - sin2x)

Important Notes on rã 2x Formula

tan2x = 2tan x / (1 − tan2x)tan2x = sin 2x/cos 2xThe derivative of tan2x is 2 sec2(2x)The integral of tan2x is (-1/2) ln |cos 2x| + C or (1/2) ln |sec 2x| + C.

☛ Related Articles:



What is Tan2x in Trigonometry?

Tan2x formula is one of the very commonly used double angle trigonometric formulas và can be expressed in terms of different trigonometric functions such as tan x, cos x, and sin x. The commonly used formula for tan2x is tan2x = 2tan x / (1 − tan2x).

What is Tan2x Formula?

Tan2x formula can be expressed in different ways such as:

tan2x = 2tan x / (1 − tan2x)tan2x = sin 2x/cos 2xtan2x = <2 cos x/(2 cos2x - 1)>√(1 - cos2x)tan2x = <2 sin x/(1 - 2 sin2x)>√(1 - sin2x)

What is the Domain & Range of Tan2x?

The domain of tan2x consists of all real numbers except those where tan2x is not defined. So, the domain of tung 2x is R - (2n + 1)π/4, n ∈ Z. The range of tan2x is all real numbers, that is, R.

How to lớn Find the Derivative of Tan2x?

The derivative of tan2x can be calculated using different methods such as the chain rule & quotient rule. The derivative of tan2x is 2 sec2(2x).

What is the Integral of Tan2x?

The integral of tan2x is given by (-1/2) ln |cos 2x| + C or (1/2) ln |sec 2x| + C.

What is the Formula of Tan2x in Terms of Cos x and Sin x?

Tan2x formula in terms of sin x and cos x is given by tan2x = 2 sin x cos x /(cos2x - sin2x)

What is rã Square x?

Tan square x is the square of the trigonometric function chảy x & is mathematically written as tan^2x. We use the tan^2x formula to solve complex integration & differentiation problems and simplify trigonometric expressions.

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What is Tan^2x Formula?

The formula for tan^2x are:

tan^2x = sec^2x - 1 ⇒ tan2x = sec2x - 1tan^2x = sin^2x / cos^2x ⇒ tan2x = sin2x/cos2xtan^2x = 1/cot^2x ⇒ tan2x = 1/cot2x

What is the Difference Between tan2x & tan^2x?

Tan2x is a double angle trigonometric formula which gives the value of the tangent function for the compound angle 2x. On the other hand, tan^2x is the whole square of the trigonometric function tanx. The value of tan2x can be both positive and negative whereas the value of tan^2x is always non-negative as the square of a number is never negative.