# Cot[Arctan(y)] = tan[Arccot(y)] = 1/y

• Alexx1
In summary, the equation "Cot[Arctan(y)] = tan[Arccot(y)] = 1/y" is a trigonometric identity that is always true for any value of y. It can be derived using the fundamental identities of trigonometry and is significant in solving trigonometric equations and simplifying expressions involving cotangent and tangent functions. However, it cannot be used to directly solve for y and is only valid for real values of y where the inverse tangent and inverse cotangent functions are defined and y is not equal to 0.
Alexx1
Hello,

can someone prove this please?

cot[Arctan(y)] = tan[Arccot(y)] = 1/y

cot(u) = 1/tan(u)

cot(arctan(y)) = 1/tan(arctan(y))

1/y

The other one's the same, with tan(u) = 1/cot(u) as the relationship.

Thx!

## 1. What does the equation "Cot[Arctan(y)] = tan[Arccot(y)] = 1/y" mean?

The equation represents a trigonometric identity, which means that it is always true for any value of y. It shows that the cotangent of the inverse tangent of y is equal to the tangent of the inverse cotangent of y, which is also equal to 1 divided by y.

## 2. How is this equation derived?

The equation can be derived using the fundamental identities of trigonometry, specifically the reciprocal identities and the definition of inverse trigonometric functions. By manipulating these identities, we can arrive at the given equation.

## 3. What is the significance of this equation?

This equation is important in solving trigonometric equations and simplifying expressions involving cotangent and tangent functions. It also helps in understanding the relationship between inverse trigonometric functions and their reciprocals.

## 4. Can this equation be used to solve for y?

No, this equation cannot be used to solve for y directly. It is an identity, not an equation that can be solved for a specific value of y. However, it can be used to simplify expressions involving cotangent and tangent functions.

## 5. Are there any restrictions on the value of y for this equation to hold true?

Yes, there are restrictions on the value of y. The equation is only valid for values of y where the inverse tangent and inverse cotangent functions are defined, which is when y is a real number and not equal to 0.

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