Question

What are the reciprocals of the trigonometric functions $\sin x,\cos x,\tan x$?

Answer 1

We should use the concept of reciprocity to find the reciprocal of the trigonometric functions. The reciprocal of any term is $\dfrac{b}{a}$ if the original term is $\dfrac{a}{b}$. It means we can define the reciprocal term as a power of -1 of the original full term or inverse of the original complete term.

Complete step by step answer:
According to our question, it is asked to determine the reciprocal of the trigonometric function $\sin x$. As we know that the reciprocal of any term is the inverse of that complete term. It means that if we take the reciprocal of the trigonometric terms, then it will be calculated by taking the whole inverse of a trigonometric function with the angle also. It means that the inverse is taken of the value of that trigonometric value if the angle is given.
Like we can understand it by an example that if an angle is given as $\dfrac{\pi }{4}$. And then we have to give the function as $\sin x=f\left( x \right)$ and $x=\dfrac{\pi }{4}$. If we take the reciprocal of $f\left( x \right)$, then,
$=\dfrac{1}{\sin \left( \dfrac{\pi }{4} \right)}=\dfrac{1}{\dfrac{1}{\sqrt{2}}}=\dfrac{\sqrt{2}}{1}=\sqrt{2}$
So, the reciprocal of $f\left( x \right)=\sin x$ at $x=\dfrac{\pi }{4}$ is $\sqrt{2}$, which is equal to the inverse of $\sin \dfrac{\pi }{4}$.
So, we can see that the inverse of trigonometric functions is equal to reciprocals at the given angles.
According to our question the reciprocal identities of trigonometric functions are:
Let $a=\sin \left( x \right),b=1$
So, if we take the reciprocal of $\sin x$, it is equal to,
$\dfrac{1}{\sin x}=\operatorname{cosec}x$
If we take the reciprocal of $\cos x$, it is equal to,
$\dfrac{1}{\cos x}=secx$
And if we take the reciprocal of $\tan x$, it is equal to,
$\dfrac{1}{\tan x}=\cot x$
So, these are the reciprocal of trigonometric functions $\sin x,\cos x,\tan x$.

Note: The reciprocal identities of trigonometric functions are used to convert one trigonometric function to another trigonometric function. But these will change in the pair of both. And if we want to change the paired trigonometric function, then we have to use the other identities of trigonometry.