2 Tan 2x Sec 2x

Tan2x Formula

Tan2x is an of import trigonometric function. Tan2x formula is ane of the very commonly used double bending trigonometric formulas and can be expressed in terms of different trigonometric functions such equally tan ten, cos 10, and sin 10. Equally nosotros know that tan x is the ratio of sine and cosine office, therefore the tan2x identity can besides be expressed as the ratio of sin 2x and cos 2x.

In this article, nosotros will learn the tan2x and tan^2x formula, its proof, and express information technology in terms of different trigonometric functions. We will also explore the graph of tan2x and its flow forth with the concept of tan square x and solve examples for a ameliorate understanding.

1.	What is Tan2x in Trigonometry?
2.	Tan2x Formula
3.	Tan2x Formula Proof
4.	Tan2x Graph
five.	Tan^2x (Tan Square 10)
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 and has a formula that is used to solve diverse bug in trigonometry. Tan2x is an important double bending formula, that is, a trigonometry formula where the angle is doubled. Information technology can be expressed in terms of tan 10 and also equally a ratio of sin2x and cos2x. Since the reciprocal of tan x is cot ten, therefore nosotros can write tan2x equally the reciprocal of cot 2x, that is, tan2x = ane/cot2x. Allow u.s.a. meet the tan2x formula:

Tan2x Formula

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

• tan2x = 2tan x / (one−tan²x)
• tan2x = sin 2x/cos 2x

Tan2x Formula Proof

Tan2x formula can be derived using two different methods. First, nosotros will use the angle addition formula for the tangent function to derive the tan2x identity. Notation that we can write the double angle 2x equally 2x = 10 + x. We will use the following trigonometric formula to evidence the formula for tan2x:

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

We accept

tan2x = tan (ten + x)

= (tan ten + tan 10)/(i - tan x tan x)

= two tan x/(i - tan²x)

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

Tan2x Identity Proof Using Sin and Cos

Now, we will derive the tan2x formula by expressing tan as a ratio of sin and cos. We will use the post-obit trigonometric formulas:

• tan x = sin 10/ cos x
• sin 2x = ii sin x cos x
• cos 2x = cos²x - sin²x

Using the in a higher place formulas, we accept

tan2x = sin 2x/cos 2x

= two sin 10 cos x/(cos²ten - sinⁱⁱx)

Divide the numerator and denominator of 2 sin ten cos x/(1 - 2 sin²x) past cos^twox

tan2x = [2 sin ten cos ten/cos²x]/[(cos^two10 - sin²x)/cos²10]

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

= 2 tan ten/(ane - tan²x)

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

Tan2x Graph

The graph of tan2x looks similar to the graph of tan 10. We know that the period of tan x is π. Since the period of tan bx is given past π/|b|, the period of tan2x is π/2. Given beneath is the graph of tan2x and as nosotros tin discover from the graph, the value of tan2x repeats afterwards every π/2 radians. As well, the value of tanx is equal to goose egg 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.

Tan^2x (Tan Square ten)

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. Equally we know that tan x can exist expressed equally the ratio of sinx and cosx, therefore we can express tan^2x can every bit the ratio of sin square x and cos square 10. We apply the tan^2x formula to solve complex integration and differentiation problems and simplify trigonometric expressions. In the next section, nosotros will derive and discuss the formula of tan square x.

Tan^2x Formula

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

• tan^2x = sec^2x - ane ⇒ tan²x = sec^two10 - one
• tan^2x = sin^2x / cos^2x ⇒ tan²x = sin²x/cos²x
• tan^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 express tan2x in terms of cos x.

• tan x = sin 10/ cos x
• sin 2x = 2 sin x cos 10
• cos 2x = 2 cos²x - i
• sin x = √(ane - cosⁱⁱ10)

Using the above formulas, we have

tan2x = sin 2x/ cos 2x

= 2 sin x cos 10/(2 cosⁱⁱten - one)

= [2 √(1 - cos²10) cos 10/(2 cos²ten - 1)]

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

tan2x = [2 sin x/(1 - 2 sin²x)]√(1 - sin²x)

Of import Notes on Tan 2x Formula

• tan2x = 2tan 10 / (1 − tan²ten)
• tan2x = sin 2x/cos 2x
• The derivative of tan2x is two sec²(2x)
• The integral of tan2x is (-1/2) ln |cos 2x| + C or (ane/2) ln |sec 2x| + C.

☛ Related Articles:

• Tangent Formulas
• Cos 2x

Tan2x Examples

1. Example 1: Decide the value of tan2x if tan x = 4/3.

   Solution: We know that tan2x = 2tan x / (1 − tan²x). If tan x = 4/3, we accept tan^twox = 16/9.

   tan2x = 2tan x / (1 − tanⁱⁱx)

   = 2(iv/three)/(1 - 16/nine)

   = (8/three)/(-vii/9)

   = -24/7

   Answer: tan2x = -24/seven

2. Example 2: Find the value of tan2x if sin 10 = 12/13 and cos ten = 5/13

   Solution: Nosotros know that tan2x = sin 2x/ cos 2x

   = 2 sin ten cos x /(cos²x - sin²x)

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

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

   = -120/119

   Answer: tan2x = -120/119

3. Case 3: Evaluate the derivative and integral of tan2x.

   Solution: We know that the derivative of trigonometric part tan x is given past secⁱⁱx. The derivative of tan2x can be calculated using dissimilar methods such equally the concatenation rule and quotient dominion. Allow usa determine the derivative of tan2x using the chain dominion.

   d(tan2x)/dx = d(tan 2x)/d(2x) × d(2x)/dx

   = sec²2x × 2

   = 2 sec²(2x)

   Now, we will determine the integral of tan2x. Nosotros know that the integral of tan ten is -ln |cos x| + C or ln |sec ten| + C. Using the formulas of integration, the integral of tan2x is given by,

   ∫ tan2x = (-1/2) ln |cos 2x| + C

   = (1/2) ln |sec 2x| + C

   Respond: Hence the derivative of tan2x is ii sec²(2x) and the integral of tan2x is given past (-1/ii) ln |cos 2x| + C or (one/two) ln |sec 2x| + C.

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FAQs on Tan2x Formula

What is Tan2x in Trigonometry?

Tan2x formula is ane of the very ordinarily used double angle trigonometric formulas and can exist expressed in terms of unlike trigonometric functions such as tan x, cos x, and sin ten. The commonly used formula for tan2x is tan2x = 2tan ten / (1 − tan^twox).

What is Tan2x Formula?

Tan2x formula can be expressed in different means such equally:

• tan2x = 2tan ten / (1 − tan²ten)
• tan2x = sin 2x/cos 2x
• tan2x = [2 cos x/(2 cos²10 - one)]√(ane - cos²10)
• tan2x = [2 sin 10/(1 - 2 sin²x)]√(1 - sin²ten)

What is the Domain and Range of Tan2x?

The domain of tan2x consists of all real numbers except those where tan2x is not defined. And then, the domain of tan 2x is R - {(2n + 1)π/iv, n ∈ Z}. The range of tan2x is all real numbers, that is, R.

How to Find the Derivative of Tan2x?

The derivative of tan2x tin be calculated using different methods such equally the chain rule and quotient dominion. The derivative of tan2x is 2 sec^two(2x).

What is the Integral of Tan2x?

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

What is the Formula of Tan2x in Terms of Cos 10 and Sin x?

Tan2x formula in terms of sin 10 and cos ten is given past tan2x = ii sin x cos x /(cos²10 - sin^twox)

What is Tan Square x?

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

What is Tan^2x Formula?

The formula for tan^2x are:

• tan^2x = sec^2x - 1 ⇒ tanⁱⁱx = sec²x - 1
• tan^2x = sin^2x / cos^2x ⇒ tan²x = sinⁱⁱx/cosⁱⁱx
• tan^2x = one/cot^2x ⇒ tan^twox = 1/cot²x

What is the Difference Between tan2x and 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 exist both positive and negative whereas the value of tan^2x is e'er non-negative as the foursquare of a number is never negative.

2 Tan 2x Sec 2x,

Source: https://www.cuemath.com/tan2x-formula/

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