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

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

Inverse of tan to inverse of sin

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

Now we can write as

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

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

Now it can be written as

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

In Right angle triangle

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

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

or

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

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

So value of the function will be in the range $[-\pi/2 , 0]$

Now we know from the property that

$tan^{-1} (-x)= – tan^{-1} (x)$

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

So we have same formula for any values of x

$tan^{-1} x = sin ^{-1} \frac {x}{\sqrt { 1 +x^2}}$

Inverse of tan to inverse of cos

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

from the above, we can write that

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

or

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

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

So value of the function will be in the range $[-\pi/2 , 0]$

Now

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

This makes sense also as Range of the cos 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 tan to inverse of cosec

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

from the above, we can write that

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

or

$tan^{-1} x = cossec^{-1} \frac {\sqrt {1+x^2}}{x}$

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

So value of the function will be in the range $[-\pi/2 , 0]$

Now

$tan^{-1} x = – tan^{-1} |x| = – cosec^{-1} \frac {\sqrt { 1 +|x|^2}}{|x|}= cosec^{-1} \frac {\sqrt { 1 +x^2}}{x}$

So we have same formula for any values of x

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

Inverse of tan to inverse of sec

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

from the above, we can write that

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

or

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

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

So value of the function will be in the range $[-\pi/2 , 0]$

Now

$tan^{-1} x = – tan^{-1} |x| = – sec^{-1} \sqrt { 1 +|x|^2}= – sec^{-1} \sqrt { 1 +x^2}$

This makes sense also as Range of the sec 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 tan to inverse of cot

This we already know from the property

$tan^{-1} x = cot^{-1} \frac {1}{x}$ if x > 0

$tan^{-1} x = cot^{-1} \frac {1}{x} – \pi $ if x < 0