Question

Find the principal value of the following: $\operatorname{cosec}\left( {{\sin }^{-1}}x+{{\cos }^{-1}}x \right)$

Answer 1

Hint: To solve this question what we will do is, firstly by using inverse trigonometric identity that is \[si{{n}^{-1}}x\text{ }+\text{ }co{{s}^{-1}}x\text{ }=~\dfrac{\pi }{2}\] where $x\in \left[ -1,1 \right]$ , we will substitute the value of \[si{{n}^{-1}}x\text{ }+\text{ }co{{s}^{-1}}x\text{ }=~\dfrac{\pi }{2}\] in equation $\operatorname{cosec}\left( {{\sin }^{-1}}x+{{\cos }^{-1}}x \right)$ and then using value of $\cos ec\left( \dfrac{\pi }{2} \right)$ which is equals to 1 , we will find out the principal value of $\operatorname{cosec}\left( {{\sin }^{-1}}x+{{\cos }^{-1}}x \right)$.

Complete step-by-step solution:
Before we solve the question, let us see what is the meaning of the principal value of inverse trigonometric functions.
Now, the principal value of the inverse trigonometric function at a point x is the value of the inverse function at a point x, which lies in the range of the principal branch.
For example, principal branch of ${{\cos }^{-1}}x$ is $[0,\pi ]$ and principal value of ${{\sin }^{-1}}x$ is $[\dfrac{-\pi}{2}, \dfrac{\pi}{2} ]$ .
Now, in question we are asked to find the principal value of $\operatorname{cosec}\left( {{\sin }^{-1}}x+{{\cos }^{-1}}x \right)$.
As, we know that, \[si{{n}^{-1}}x\text{ }+\text{ }co{{s}^{-1}}x\text{ }=~\dfrac{\pi }{2}\] where $x\in \left[ -1,1 \right]$
So, substituting value of \[si{{n}^{-1}}x\text{ }+\text{ }co{{s}^{-1}}x\text{ }=~\dfrac{\pi }{2}\] in $\operatorname{cosec}\left( {{\sin }^{-1}}x+{{\cos }^{-1}}x \right)$, we get
$\operatorname{cosec}\left( si{{n}^{-1}}x\text{ }+\text{ }co{{s}^{-1}}x\text{ }=~\dfrac{\pi }{2} \right)$
$=\operatorname{cosec}\left( \dfrac{\pi }{2} \right)$
Now, we know that at $x=\left( \dfrac{\pi }{2} \right)$, value of function $\operatorname{cosecx}$ is 1.
So, we get $\operatorname{cosec}\left( \dfrac{\pi }{2} \right)=1$
Hence, the principal value of $\operatorname{cosec}\left( si{{n}^{-1}}x+co{{s}^{-1}}x \right)$ is 1.

Note: To solve the questions of inverse trigonometric questions, one must know the meaning of principal value of inverse trigonometric function and principal branch too. Also, one must know the following inverse trigonometric formulas which are \[si{{n}^{-1}}x\text{ }+\text{ }co{{s}^{-1}}x\text{ }=~\dfrac{\pi }{2}\] for $x\in \left[ -1,1 \right]$, \[{{\tan }^{-1}}x\text{ }+\text{ }{{\cot }^{-1}}x\text{ }=~\dfrac{\pi }{2}\] for \[x\in R\] and \[{{\sec }^{-1}}x\text{ }+\text{ }{{\operatorname{cosec}}^{-1}}x\text{ }=~\dfrac{\pi }{2}\] for |x| ≥ 1. Also, remember that the value of trigonometric function cosec x at $x=\left( \dfrac{\pi }{2} \right)$ is equal to 1. Try to avoid calculation error while solving the question.