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Cot Inverse x
Cot inverse x is one of the main six inverse trigonometric functions. It is also known by different names such as arc cot x, inverse cot, and inverse cotangent. Cot inverse x is the inverse function of the trigonometric function cot x and is written as cot^{1}x. Please note that inverse cot is not the reciprocal of the cotangent function. Cot inverse x gives the measure of the angle in a rightangled triangle corresponding to the given ratio of base and perpendicular.
Let us explore the inverse trigonometric function cot inverse x, its formula, domain and range, derivative and integral, and graph. We will also solve some examples based on inverse cot for a better understanding.
1.  What is Cot Inverse x? 
2.  Cot Inverse x Formula 
3.  Domain, Range and Graph of Inverse Cot 
4.  Derivative and Integral of Cot Inverse x 
5.  FAQs on Cot Inverse x 
What is Cot Inverse x?
Cot Inverse x is an inverse trigonometric function that gives the measure of the angle in radians or degrees corresponding to the value of x. Mathematically, it is written as cot^{1}x or arccot x, pronounced as 'cot inverse x' and ' arc cot x', respectively. If a function f is invertible and its inverse is f^{1}, then we have f(x) = y ⇒ x = f^{1}(y). Therefore, we can have cot inverse x, if x = cot y, then we have y = cot^{1}x. The functioning of cot inverse x is as follows:
 If cot π/2 = 0, then cot^{1}0 = π/2
 If cot π/6 = √3, then cot^{1}√3 = π/6
 If cot π/3 = 1/√3, then cot^{1}1/√3 = π/3
 If cot π/4 = 1, then cot^{1}1 = π/4
Cot Inverse x Formula
We know that in a rightangled triangle, we have cot θ as the ratio of the adjacent side and opposite side, i.e., cot θ = Adjacent Side/Opposite Side. So, using the cot inverse x definition, we have θ = cot inverse of Adjacent Side/Opposite Side, that is, the formula for the inverse cot is:
θ = cot^{1}(Adjacent Side/Opposite Side)
Domain, Range and Graph of Inverse Cot
A function f is invertible if and only if it is bijective and the domain and range of f becomes the range and domain, respectively, of the inverse function. We know that the domain of cot x is R  nπ and the range is R, where R is the set of all real numbers and n is an integer (Because cot x is not defined for integral multiples of π). But cot x is not oneone on R  nπ, and hence it is not bijective. So, for cot x to be oneone, we can restrict the domain R  nπ of cot x to (0, π), (π, 2π), (π, 0), etc. Generally, we restrict the domain of cot x to (0, π) for it to be oneone, and hence bijective, i.e., cot x: (0, π) → R.
Hence, the domain of cot inverse x is R and its range is (0, π), i.e., cot^{1}x: R → (0, π).
Now, seeing the domain and range of inverse cot, we will plot the graph of cot inverse x.
Derivative and Integral of Cot Inverse x
Now, we will calculate the derivative and integral of the inverse cot. First, to determine the derivative, we will use trigonometric formulas. Assume y = cot^{1}x ⇒ cot y = x. Now, differentiate cot y = x w.r.t. x. We have
cosec^{2}y dy/dx = 1
⇒ dy/dx = 1/cosec^{2}y
⇒ dy/dx = 1/(1 + cot^{2}y) (Because 1 + cot^{2}θ = cosec^{2}θ)
⇒ dy/dx = 1/(1 + x^{2})
Hence the derivative of cot inverse x is 1/(1 + x^{2})
Next, we will calculate the integral of cot inverse x using the integration by parts method. We can write cot^{1}x can be written as cot^{1}x = cot^{1}x.1. Therefore, we have
∫ cot^{1}x dx = ∫ cot^{1}x.1 dx
= cot^{1}x ∫1 dx  ∫[d(cot^{1}x)/dx . ∫ 1dx] dx
= x cot^{1}x  ∫(1/(1 + x^{2})) x dx
= x cot^{1}x + ∫x/(1 + x^{2}) dx
= x cot^{1}x + I_{1}, where I_{1 }= ∫x/(1 + x^{2}) dx
Now, we will solve I_{1. }Assume that 1 + x^{2} = u. Then 2x dx = du (or) x dx = du/2. We have,
∫x/(1 + x^{2}) dx = ∫(1/u) (1/2) du
= (1/2) lnu + C
= (1/2) ln 1 + x^{2} + C
Therefore, we have
∫ cot^{1}x dx = x cot^{1}x + ∫x/(1 + x^{2}) dx
= x cot^{1}x + (1/2) ln 1 + x^{2} + C
Hence, the integral of cot inverse x is x cot^{1}x + (1/2) ln 1 + x^{2} + C.
Important Notes on Cot Inverse x
 Domain of cot^{1}x = Real numbers
 Range of cot^{1}x = (0, π)
 Derivative of inverse cot = 1/(1 + x^{2})
 Cot Inverse x integral = x cot^{1}x + (1/2) ln 1 + x^{2} + C
 Cot of Cot Inverse x is x for any x.
 Cot Inverse of cot x is x only when x is in the interval (0, π).
Related Topics on Inverse Cot
Cot Inverse x Examples

Example 1: Prove the cot inverse identity cot^{1}x + cot^{1}y = cot^{1} [(xy  1)/(y + x)]
Solution: Assume cot^{1}x = a and cot^{1}y = b, then we have cot a = x and cot b = y.
We know that cot (a + b) = (cot a cot b  1)/(cot b + cot a)
⇒ cot (a + b) = (xy  1)/(y + x)
⇒ a + b = cot^{1} [(xy  1)/(y + x)]
⇒ cot^{1}x + cot^{1}y = cot^{1} [(xy  1)/(y + x)]
Answer: Hence we have proved cot^{1}x + cot^{1}y = cot^{1} [(xy  1)/(y + x)]

Example 2: If the length of the adjacent side and opposite side is 4 units in arightangled triangle, determine the measure of the angle in consideration.
Solution: We know that cot θ = Adjacent Side/Opposite Side
⇒ θ = cot^{1}(Adjacent Side/Opposite Side)
⇒ θ = cot^{1}(4/4)
⇒ θ = cot^{1}1
⇒ θ = 45°
Answer: Hence the angle in consideration is 45°.
FAQs on Cot Inverse x
What is Cot Inverse x in Inverse Trigonometry?
Cot Inverse x is an inverse trigonometric function that gives the measure of the angle in radians or degrees corresponding to the value of x. Mathematically, it is written as cot^{1}x or arccot x
What is the Inverse Cot Formula?
We have cot θ as the ratio of the adjacent side and opposite side, i.e., cot θ = Adjacent Side/Opposite Side. Therefore, we have θ = cot inverse of Adjacent Side/Opposite Side, that is, the formula for the inverse cot is θ = cot^{1}(Adjacent Side/Opposite Side).
How to Find the Domain of Cot Inverse x?
The range of cot x is R, therefore the domain of inverse cot is R, i.e., all real numbers.
What is the Range of Cot Inverse x?
The domain of a function becomes the range of its inverse. The domain of cot x is R  nπ but cot x is not oneone on R  nπ, and hence it is not bijective. So, for cot x to be oneone, we restrict the domain R  nπ of cot x to (0, π). Therefore, the range of cot inverse x is (0, π).
What is the Derivative of Cot Inverse x?
The derivative of cot inverse x is 1/(1 + x^{2}) which can be calculated using implicit differentiation.
How Do you Find the Cot Inverse x Integral?
We can calculate the integral of cot inverse x using the integration by parts method. The integral of cot inverse x is x cot^{1}x + (1/2) ln 1 + x^{2} + C.
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