Question

How do you solve ${{\csc }^{2}}x-2=0$?

Answer 1

Hint: We have a trigonometric equation in the expression, we will simplify the equation to get the value or the range of the possible values of $x$. We will convert the cosec function to sine function and then use the general solution of sine function to get the answer.

Complete step by step answer:
We have the given expression given to us as ${{\csc }^{2}}x-2=0$
On rearranging the equation, we get:
$\Rightarrow {{\csc }^{2}}x=2$
Now on taking the square root on both the sides we get:
$\Rightarrow \csc x=\pm \sqrt{2}$
Now on using the trigonometric inverse function we can write the equation as:
$\Rightarrow x={{\csc }^{-1}}\left( \pm \sqrt{2} \right)$
Now we know about the trigonometric identity that ${{\csc }^{-1}}x={{\sin }^{-1}}\dfrac{1}{x}$
On substituting the value in the equation, we get:
\[\Rightarrow x={{\sin }^{-1}}\left( \pm \dfrac{1}{\sqrt{2}} \right)\]
We know that ${{\sin }^{-1}}\left( -x \right)=-{{\sin }^{-1}}\left( x \right)$and since we know that the principal value of ${{\sin }^{-1}}\left( \dfrac{1}{\sqrt{2}} \right)$is $\dfrac{\pi }{4}$ and $\dfrac{3\pi }{4}$ radians
The principal value of ${{\sin }^{-1}}\left( -\dfrac{1}{\sqrt{2}} \right)$will be $-\dfrac{\pi }{4}$ and $-\dfrac{3\pi }{4}$ radians
And since the value of ${{\sin }^{-1}}\left( x \right)$will be the same after every $2\pi $ radians, the solution of the expression in the general format will be as:
$S=\left\{ \pm \dfrac{\pi }{4}+2\pi n,\pm \dfrac{3\pi }{4}+2\pi n \right\}$, where $n$ is any integer.
The set $S$ is the required solution.

Note: The formula used over here is for $\sin (n\pi +x)$ ,
It is to be remembered that $\sin (n\pi +x)={{(-1)}^{n}}\sin x$
Basic trigonometric formulas should be remembered to solve these types of sums.
The inverse trigonometric function of $\sin x$ which is ${{\sin }^{-1}}x$ used in this sum.
For example, if $\sin x=a$ then $x={{\sin }^{-1}}a$ .
And ${{\sin }^{-1}}(\sin x)=x$ is a property of the inverse function.
There also exists inverse functions for the other trigonometric relations such as cos and tan.
The inverse function is used to find the angle $x$ from the value of the trigonometric relation.
The trigonometric function given to us in the question is $\csc x$ which is actually pronounced as cosec but written as csc. It is the reciprocal function of sine.