Question

How do you find the value of $\tan ({{\csc }^{-1}}(2))$ ?

Answer 1

In this question, we have to find the value of the trigonometric function. So, we will use the trigonometric formula to get the required result. Therefore, we start our question by first solving the brackets of the brackets, that is we first find the value of ${{\csc }^{-1}}(2)$by getting the angle value of cosec inverse at 2. Then, we will find the value of the tan function at that angle, which is our required result to the problem.

Complete step-by-step answer:
According to the question, we have to find the value of a trigonometric function.
So, we will use the trigonometric formula.
The trigonometric function given to us is $\tan ({{\csc }^{-1}}(2))$ ------ (1)
We first solve the trigonometric function ${{\csc }^{-1}}(2)$. As we know, the value of the inverse trigonometric function is equal to the angle where the value lie, therefore the angle value of cosec inverse at 2 is equal to
${{\csc }^{-1}}(2)=\dfrac{\pi }{6}$ ------------- (2)
Now, we will put the value of equation (2) in equation (1), we get
$\tan \left( \dfrac{\pi }{6} \right)$
Now, we know that from the trigonometric table, the value of the tan function at $\dfrac{\pi }{6}$ is equal to
$\dfrac{1}{\sqrt{3}}$ which is our required solution.
Therefore, for the trigonometric equation $\tan ({{\csc }^{-1}}(2))$, its simplified value is equal to $\dfrac{1}{\sqrt{3}}$ .

Note: While solving this question, do mention how you are using the trigonometric formula in that step, to avoid confusion and you will get an accurate answer. You can also find the angle value of the cosec inverse function, by using the trigonometric formula $\text{cose}{{\text{c}}^{-1}}\text{(x)}=\left( \dfrac{1}{{{\sin }^{-1}}x} \right)$ , that is find the value of the sin inverse function and take the reciprocal, to get the required result to the problem.