Question

The value of the expression ${{\tan }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)=$
(A) $\dfrac{5\pi }{4}$
(B) $\dfrac{\pi }{4}$
(C) $-\dfrac{\pi }{4}$
(D) None of these

Answer 1

Hint: We solve this question by first considering the given expression ${{\tan }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)$. Then we substitute the value of $\tan \dfrac{3\pi }{4}$ in it. Then we consider the formula, ${{\tan }^{-1}}\left( -x \right)=-{{\tan }^{-1}}\left( x \right)$. Then we use it to simplify the value of a given expression. Then we use the principle value of the value present inside the ${{\tan }^{-1}}$ function present in the obtained value. Then we use the formula, ${{\tan }^{-1}}\left( \tan x \right)=x\ \ \ for\ \ x\in \left( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right)$ and substitute this value in the above obtained value and find the required value.

Complete step-by-step solution:
The expression we are given is ${{\tan }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)$.
First, let us substitute the value of $\tan \dfrac{3\pi }{4}$ in the given expression.
As we know that $\tan \dfrac{3\pi }{4}=-1$, the expression we have is converted as,
$\Rightarrow {{\tan }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)={{\tan }^{-1}}\left( -1 \right)$
Now let us consider the formula,
${{\tan }^{-1}}\left( -x \right)=-{{\tan }^{-1}}\left( x \right)$ for all $x\in R$
So, using it we can write the above equation as,
$\Rightarrow {{\tan }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)=-{{\tan }^{-1}}\left( 1 \right)............\left( 1 \right)$
Now let us first find the value of ${{\tan }^{-1}}\left( 1 \right)$.
We know that, $\tan \dfrac{\pi }{4}=1$.
So, we can write ${{\tan }^{-1}}\left( 1 \right)$ as,
$\Rightarrow {{\tan }^{-1}}\left( 1 \right)={{\tan }^{-1}}\left( \tan \dfrac{\pi }{4} \right)$
Substituting this value in equation (1) we get,
$\Rightarrow {{\tan }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)=-{{\tan }^{-1}}\left( \tan \dfrac{\pi }{4} \right)$
Now let us consider the formula,
${{\tan }^{-1}}\left( \tan x \right)=x\ \ \ for\ \ x\in \left( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right)$
As $\dfrac{\pi }{4}\in \left( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right)$, we get value in the above equation as,
$\Rightarrow {{\tan }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)=-\dfrac{\pi }{4}$
So, we get the value of ${{\tan }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)$ as $-\dfrac{\pi }{4}$.
Hence the answer is Option C.

Note: The common mistake one makes while solving this problem is that one might assume the value of given expression as $\theta $ and solve it as,
$\begin{align}
  & \Rightarrow {{\tan }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)=\theta \\
 & \Rightarrow \tan \theta =\tan \dfrac{3\pi }{4} \\
 & \Rightarrow \theta =\dfrac{3\pi }{4} \\
\end{align}$
Then by checking with the options one might mark the answer as Option D.
But it is wrong as the range of ${{\tan }^{-1}}x$ is $\left( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right)$. So, we need to consider the principal value of $\theta $ above while solving.