how to convert cot inverse to sin, cos, tan, sec, cosec x inverse

We often need to convert inverse of cot to either inverse of cos, sin, tan, sec, cosec x .In this post we will see how to convert cot inverse to sin, cos, tan, sec, cosec x inverse

Inverse of cot to inverse of sin

Case 1
$cot^{-1} x$ and x > 0

Now we can write as

$\theta=cot^{-1} x$
$cot \theta =x$

Now we know that here $\theta \in [0,\pi/2]$, so it is an acute angle

Now it can be written as

$cot \theta =\frac {x}{1} = \frac {base}{perp}$

In Right angle triangle

Now then hyp becomes
$\text{hyp} = \sqrt { 1 +x^2}$

So , $sin \theta = \frac {perp}{hyp} = \frac {1}{\sqrt { 1 +x^2}}$

or

$\theta= sin^{-1} \frac {1}{\sqrt { 1 +x^2}}$
or $cot^{-1} x = sin ^{-1} \frac {1}{\sqrt { 1 +x^2}}$

Case 2
$cot^{-1} x$ and x < 0

So value of the function will be in the range $[\pi/2 , \pi)$

Now we know from the property that

$cot^{-1} (-x)= \pi – cot^{-1} (x)$

This can be written as
$cot^{-1} x = \pi – cot^{-1} |x| =\pi – sin ^{-1} \frac {1}{\sqrt { 1 +x^2}}$

This makes sense also as Range of the cot and sin function differ. We can convert with out worrying about the sign in $[0, \pi/2]$ as it is common

Thus ,we have different formula depending on the values of x

Inverse of cot to inverse of cos

Case 1
$cot^{-1} x$ and x > 0

from the above, we can write that

$cos \theta = \frac {base}{hyp} = \frac {x}{\sqrt {1+x^2}}$

or

$cot^{-1} x = cos^{-1} \frac {x}{\sqrt { 1 +x^2}}$

Case 2
$cot^{-1} x$ and x < 0

So value of the function will be in the range $[\pi/2 , \pi)$

Now we know from the property that

$cot^{-1} (-x)= \pi – cot^{-1} (x)$

$cot^{-1} x = \pi – cot^{-1} |x| = \pi – cos^{-1} \frac {|x|}{\sqrt { 1 +|x|^2}}= cos^{-1} \frac {x}{\sqrt { 1 +x^2}}$

So we have same formula for any values of x

$cot^{-1} x = cos^{-1} \frac {x}{\sqrt { 1 +x^2}}$

Inverse of cot to inverse of tan

Case 1
$cot^{-1} x$ and x > 0

from the above, we can write that

$tan \theta = \frac {perp}{base} = \frac {1}{x}$

or

$cot^{-1} x = tan^{-1} \frac {1}{x}$

Case 2
$cot^{-1} x$ and x < 0

Now we know from the property that

$cot^{-1} (-x)= \pi – cot^{-1} (x)$

$cot^{-1} x = \pi – cot^{-1} |x| = \pi – tan^{-1} \frac {1}{|x|}= \pi + tan^{-1} \frac {1}{x}$

This makes sense also as Range of the cot and tan function differ. We can convert with out worrying about the sign in $[0, \pi/2]$ as it is common

Thus ,we have different formula depending on the values of x

Inverse of cot to inverse of sec

Case 1
$cot^{-1} x$ and x > 0

from the above, we can write that

$sec \theta = \frac {hyp}{base} = \frac {\sqrt {1+x^2}}{x}$

or

$cot^{-1} x = sec^{-1} \frac {\sqrt {1+x^2}}{x}$

Case 2
$cot^{-1} x$ and x < 0

So value of the function will be in the range $[\pi/2 , \pi)$

Now we know from the property that

$cot^{-1} (-x)= \pi – cot^{-1} (x)$

$cot^{-1} x = \pi – cot^{-1} |x| = \pi – sec^{-1} \frac {\sqrt {1+x^2}}{|x|}= sec^{-1} \frac {\sqrt {1+x^2}}{x}$

So we have same formula for any values of x

$cot^{-1} x = sec^{-1} \frac {\sqrt {1+x^2}}{x}$

Inverse of cot to inverse of cosec

Case 1
$cot^{-1} x$ and x > 0

from the above, we can write that

$sec \theta = \frac {hyp}{perp} =\sqrt {1+x^2}$

or

$cot^{-1} x = cosec^{-1} \sqrt {1+x^2}$

Case 2
$cot^{-1} x$ and x < 0

So value of the function will be in the range $[\pi/2 , \pi)$

Now we know from the property that

$cot^{-1} (-x)= \pi – cot^{-1} (x)$

$cot^{-1} x = \pi – cot^{-1} |x| = \pi – cosec^{-1} \sqrt {1+x^2}$

I hope you like this article on how to convert cot inverse to sin, cos, tan, sec, cosec x inverse

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