# Thread: inverse trignometric functions in respect of Cotangent inverse

1. ## inverse trignometric functions in respect of Cotangent inverse

Sir,

To find the value of cot -1 (-x)
I proceeded like this,
Let Cot -1 (-x) = y
-x = Cot y
x = - Cot y = Cot ( –y)
Cot -1 x = - y = Cot -1 (-x) .

is Cot -1 x = Cot -1 (-x).

To find the value of cot -1 (-x)

My friend said as follow :
Let Cot -1 (-x) = y
-x = Cot y
x = - Cot y = Cot ( π–y)
Cot -1 x = π - y = π - Cot -1 (-x)
Cot -1 x = π - Cot -1 (-x).

Whether both the results are correct?

Which one I have to adopt and why?

Kindly enlighten me.

with warm regards,

Aranga

2. ## Re: inverse trignometric functions in respect of Cotangent inverse

Originally Posted by arangu1508
Sir,

To find the value of cot -1 (-x)
I proceeded like this,
Let Cot -1 (-x) = y
-x = Cot y
x = - Cot y = Cot ( –y)
Cot -1 x = - y = Cot -1 (-x) .

is Cot -1 x = Cot -1 (-x).
Not quite unless $y=-y=0$.

You have $\text{Cot}^{-1} x = -y \Longrightarrow y = -\text{Cot}^{-1} x$ and $y = \text{Cot}^{-1}(-x)$. This gives you $\text{Cot}^{-1}(-x) = -\text{Cot}^{-1}x$. So, arccotangent is an odd function.

Now, your friend's approach is not correct. This is because $\text{Cot}^{-1}$ returns a value between $-\dfrac{\pi}{2}$ and $\dfrac{\pi}{2}$. So, $\pi - y$ will yield a value outside the range of the $\text{Cot}^{-1}$ function. Note that this is different from $\cot^{-1}$. The latter is the notation used for the generic inverse of cotangent, which has an infinite number of values that satisfy it. In fact, $\cot^{-1}(-x) = n\pi - \cot^{-1}x$ is true for any integer $n$. But, for the $\text{Cot}^{-1}$ function, that is the notation for a specific value of the arccotangent expression that falls between $-\dfrac{\pi}{2}$ and $\dfrac{\pi}{2}$.

Does that make sense?

3. ## Re: inverse trignometric functions in respect of Cotangent inverse

thank you it is very useful.