Question

How to simplify ${\displaystyle \frac{\csc \left(-x\right)}{\cot \left(-x\right)}}$ ?

Answer 1

Hint: csc stands for cosec. $\cos ec\left(x\right)$ is a trigonometric function and is reciprocal of $\sin \left(x\right)$ . $\cos ec\left(x\right)$ can be defined as the ratio between hypotenuse and perpendicular to theta ( $\theta$ ). $\cot \left(x\right)$ is also a trigonometric function and is reciprocal to $\tan \left(x\right)$ . $\cot \left(x\right)$ can be defined as the ratio between base and perpendicular to theta ( $\theta$ ). Both of these trigonometric functions are odd functions.

Complete step-by-step answer:
As we already know that both $\cos ec\left(x\right)$ and $\cot \left(x\right)$ are odd functions, which means that $f\left(-x\right)=-f\left(x\right)$ . So, we can say that,
 $\Rightarrow \csc \left(-x\right)=-\csc \left(x\right)$ ---(1)
 $\Rightarrow \cot \left(-x\right)=-\cot \left(x\right)$ ---(2)
We are given that ${\displaystyle \frac{\csc \left(-x\right)}{\cot \left(-x\right)}}$ ,
Using equation (1) and (2) in given question we get,
 $\Rightarrow {\displaystyle \frac{\csc \left(x\right)}{\cot \left(x\right)}}$ ---(3)
We know that $\cos ec\left(x\right)$ is reciprocal of $\sin \left(x\right)$ which means
 $\Rightarrow \csc \left(x\right)={\displaystyle \frac{1}{\sin \left(x\right)}}$ ---(4)
And $\cot \left(x\right)$ is the reciprocal of $\tan \left(x\right)$ which means
 $\Rightarrow \cot \left(x\right)={\displaystyle \frac{\cos \left(x\right)}{\sin \left(x\right)}}$ ---(5)
Now, substituting the values of equation (4) and (5) in equation (3), we get,
 $$\begin{gathered} \Rightarrow {\displaystyle \frac{{\displaystyle \frac{1}{\sin \left(x\right)}}}{{\displaystyle \frac{\cos \left(x\right)}{\sin \left(x\right)}}}} \\ \Rightarrow {\displaystyle \frac{1}{\sin \left(x\right)}}\times {\displaystyle \frac{\sin \left(x\right)}{\cos \left(x\right)}} \end{gathered}$$
 $\sin \left(x\right)$ cuts and cancels off and we get,
 $\Rightarrow {\displaystyle \frac{1}{\cos \left(x\right)}}$ ---(6)
We know that, $\sec \left(x\right)$ is the reciprocal of $\cos \left(x\right)$ which means that, $\sec \left(x\right)={\displaystyle \frac{1}{\cos \left(x\right)}}$ . Substituting the same in equation (6) and we get,
 $\Rightarrow \sec \left(x\right)$
Thus, ${\displaystyle \frac{\csc \left(-x\right)}{\cot \left(-x\right)}}=\sec \left(x\right)$ .
So, the correct answer is “Sec x”.

Note: Before solving any trigonometric question, students should keep all the necessary trigonometric identities in their mind. This will help them to solve any question easily. Students should also remember that $\cos ec\left(x\right)$ is not the inverse of $\sin \left(x\right)$ and cannot be written as ${\sin }^{-1}x$ . The same applies to $\sec \left(x\right)$ as well as $\cot \left(x\right)$ . Students should keep in mind that $\sec \left(x\right)$ is the reciprocal of $\cos \left(x\right)$ , $\cos ec\left(x\right)$ is the reciprocal of $\sin \left(x\right)$ and $\cot \left(x\right)$ is the reciprocal of $\tan \left(x\right)$ . In addition to $\cos ec\left(x\right)$ and $ \cot \le