Question

Prove the following:
 $\dfrac{{\csc x}}{{\sec x}} = \cot x$

Answer 1

Hint: The given trigonometric is $\dfrac{{\csc x}}{{\sec x}}$. We have to prove that it is equal to $\cot x$.
We use trigonometric function to solve this question such as $\csc x = \dfrac{1}{{\sin x}}$, $\sec x = \dfrac{1}{{\cos x}}$ and $\cot x = \dfrac{{\cos x}}{{\sin x}}$. After putting the value of $\csc x$ and $\sec x$ we will get our result.

Complete step by step answer:
We have,
$\dfrac{{\csc x}}{{\sec x}} = \cot x$
Let’s take the left side of the equation
$ \Rightarrow \dfrac{{\csc x}}{{\sec x}}$
We know that
$\csc x = \dfrac{1}{{\sin x}}$ and $\sec x = \dfrac{1}{{\cos x}}$
Put the value of $\csc x$ and $\sec x$. We get,
$ \Rightarrow \dfrac{{\csc x}}{{\sec x}} = \dfrac{{\dfrac{1}{{\sin x}}}}{{\dfrac{1}{{\cos x}}}}$
Then the division we write in the form of
$\dfrac{{\dfrac{p}{q}}}{{\dfrac{r}{s}}} = \dfrac{p}{q} \times \dfrac{s}{r}$
Hence, we get,
$\dfrac{1}{{\sin x}} \times \dfrac{{\cos x}}{1}$
We rewrite the form, hence, we get
$\dfrac{{\cos x}}{{\sin x}} = \cot x$
Hence, the left side of the equation is equal to the right side of the equation i.e., $\dfrac{{\csc x}}{{\sec x}} = \cot x$
Hence proved.

Note:
We can also prove this question by taking right side of the equation i.e., $\cot x$ and write it in the term of $\sin x$ and $\cos x$ and then write $\sin x$ and $\cos x$ in term of $\csc x$ and $\sec x$. We will get the same result.