Tan3x
Tan3x is one of the triple angle identities in trigonometry. It is an important trigonometric identity that is used to solve various trigonometric and integration problems. Tan3x formula is given by tan3x = (3 tan x  tan^{3}x) / (1  3 tan^{2}x) and it can be derived using angle sum formula of tan function. Tan3x can also be expressed in terms of sin and cos as tan3x = sin 3x/cos 3x.
In this article, we will explore the concept of the tan3x formula, its application, and proof. We will also solve some examples based on tan3x for a better understanding of the tan3x identity.
1.  What is Tan3x? 
2.  Tan3x Formula 
3.  Proof of Tan3x Formula 
4.  Tan3x Differentiation and Integration 
5.  FAQs on Tan3x 
What is Tan3x?
Tan3x is a trigonometric function that gives the value of the tan function for a triple angle. The graph of tan3x is narrower than the graph of tan x. We know that the period of tan x is π radians and the period of tan bx is given by π/b. Hence the period of tan3x is π/3 radians. So, the value of tan3x repeats after every π/3 radians, that is, tan3x = tan (3x + π/3). The formula for tan3x can be derived using the tan (a + b) formula.
Tan3x Formula
Tan3x formula is an important trigonometric formula given as tan3x = (3 tan x  tan^{3}x) / (1  3 tan^{2}x) that is used to solve various mathematical problems and complex integrations. The formula for tan3x can also be written as tan3x = sin 3x/cos 3x as tangent function is a ratio of the sine function and cosine function.
Proof of Tan3x Formula
As we have studied, we know that the formula for tan3x is (3 tan x  tan^{3}x) / (1  3 tan^{2}x). Now, we will prove this formula using the angle sum formula for tan and tan 2x formula. Note that we can write 3x as 3x = 3x + x. Also, we will use the following formulas to prove that tan3x = (3 tan x  tan^{3}x) / (1  3 tan^{2}x):
Let us represent tan(3x) as follows,
⇒ tan(3x) = tan (2x + x)
Using the above trigonometric formula for tan (A + B), we have
tan (2x + x) = [tan(2x) + tan(x)] / [1  tan(2x) tan(x)]
tan(2x + x) = [tan x + {(2 tan x) / (1  tan^{2}x)}] / [1  {(2 tan x) / (1  tan^{2}x)} tan x] {Substituing the value tan 2x = (2 tan x) / (1  tan^{2}x)}
On solving, we have
tan(2x + x) = (tan x  tan^{3}x + 2 tan x) / (1  tan^{2}x  2 tan^{2}x)
= (3 tan x  tan^{3}x) / (1 3 tan^{2}x)
Thus, tan3x = (3 tan x  tan^{3}x) / (1 3 tan^{2}x)
Hence we have derived the formula for tan3x.
Tan3x Differentiation and Integration
Next, we will determine the derivative and integral of tan3x. First, let us differentiate tan3x with respect to x using the chain rule method. We know that the derivative of tan x is sec^{2}x and the derivative of ax is a, where a is a constant. Using this, we have
d(tan3x)/dx = d(tan3x)/d(3x) × d(3x)/dx
= sec^{2}(3x) × 3
= 3 sec^{2}(3x)
Therefore, the derivative of tan 3x is 3 sec^{2}(3x).
Next, for tan3x integration, we will express tan 3x as a ratio of sin 3x and cos 3x, that is, tan 3x = sin 3x/cos 3x. Also, we will use the fact that the derivative of cos 3x is 3 sin 3x. Using these facts and formulas, we have
∫tan3x dx = ∫(sin 3x / cos 3x) dx
= ∫(3 sin 3x / 3 cos 3x) dx [Multiplying the numerator and denominator by 3]
Assume cos 3x = u ⇒ 3 sin 3x = du ⇒ 3 sin 3x dx = du. We have
∫tan 3x dx = ∫(3 sin 3x / 3 cos 3x) dx
= (1/3) ∫ (1/u) du
= (1/3) ln u du
= (1/3) ln cos 3x + C
= (1/3) ln sec 3x + C
Thus, the tan 3x integration is (1/3) ln cos 3x + C or (1/3) ln sec 3x + C.
Important Notes on Tan3x
 The formula for tan3x is tan 3x = (3 tan x  tan^{3}x) / (1 3 tan^{2}x).
 The derivative of tan3x is 3 sec^{2}(3x).
 The integral of tan3x is (1/3) ln cos 3x + C or (1/3) ln sec 3x + C.
☛ Related Topics:
Examples Using Tan3x

Example 1: Calculate the value of tan 135° using tan3x formula.
Solution: Assume 3x = 135° ⇒ x = 45°. We know that tan 45° = 1. Using tan3x formula, we have
tan 135° = (3 tan 45°  tan^{3}45°) / (1 3 tan^{2}45°)
= (3 × 1  1^{3})/(1  3 × 1^{2})
= (3  1)/(1  3)
= 2/(2)
= 1
Answer: tan 135° = 1

Example 2: Prove that tan 180° is equal to 0 using tan3x formula.
Solution: Assume 3x = 180° ⇒ x = 60°. We know that tan 60° = √3. Using tan3x formula, we have
tan 180° = (3 tan 60°  tan^{3}60°) / (1 3 tan^{2}60°)
= (3 × √3  √3^{3})/(1  3 × √3^{2})
= (3√3  3√3)/(1  9)
= 0/(8)
= 0
Answer: Hence tan 180° is equal to 0

Example 3: Show that tan3x tan2x tanx = tan3x  tan2x  tanx.
Solution: Using the formula of tan (a + b), we can write tan3x as,
tan3x = tan(2x+x)
⇒ tan3x = [tan2x + tanx]/[1  tan2x tanx]  [Because tan (A + B) = (tan A + tan B) / (1  tan A tan B)]
⇒ tan3x × [1  tan2x tanx] = [tan2x + tanx]
⇒ tan3x  tan3x tan2x tanx = tan2x + tanx
⇒  tan3x tan2x tanx = tan2x + tanx  tan3x
⇒ tan3x tan2x tanx = tan3x  tan2x  tanx
Answer: Hence, we have proved the given tan3x identity.
FAQs on Tan3x
What is Tan3x Formula in Trigonometry?
The formula for tan3x is given by tan3x = (3 tan x  tan^{3}x) / (1 3 tan^{2}x). It can also be written as tan 3x = sin 3x/cos 3x.
How to Find Tan3x Formula?
Tan3x formula can be derived using the formulas tan (A + B) = (tan A + tan B) / (1  tan A tan B) and tan 2x = (2 tan x) / (1  tan^{2}x) and substituting 3x = 2x + x.
What is the Domain and Range of Tan3x?
We know that the domain of tan x is all real numbers except nπ + π/2 and the range of tan x is all real numbers. So, the domain of tan3x is all real numbers except (1/3)(nπ + π/2) = π/6 + nπ/3, where n is an integer and the range of tan3x is all real numbers.
How Do You Find the Derivative of Tan3x?
The derivative of tan3x is 3 sec^{2}(3x). It can be calculated using the chain rule method. Tan3x differentiation can also be done using the quotient rule on tan 3x = sin 3x/cos 3x.
What is Tan3x Integration?
The integral of tan3x is (1/3) ln cos 3x + C. It is also equal to (1/3) ln sec 3x + C using the properties of the logarithmic function.
What is the Period of Tan3x?
The period of tan bx is given by π/b. Therefore, the period of tan3x function is π/3 and the period of tan x is π.
What is Tan^3x?
Tan^3x is a trigonometric function that gives the value of the whole cube of the tangent function. It is called tan cube x and its formula can be determined using the tan3x formula. On simplifying the tan3x formula, the formula for tan^3x is 3tanx  tan3x (1  3tan^2x).
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