Question

How do you simplify the expression $\dfrac{1}{{\cot x}}?$

Answer 1

Hint: Here, these types of questions are very easy to solve. All you need to do is write all basic trigonometric formulas. After that relate them with each other. Like here we will write tan, sin, cosec and many basic functions of trigonometrics and After that we will relate them with each other. After simplifying we get our answer.

Formula used:
For solving this question we are using some basic trigonometric formulas.
$\begin{gathered}
   \Rightarrow \tan x = \dfrac{{\sin x}}{{\cos x}} \\
   \Rightarrow \dfrac{1}{{\cos x}} = \sec x \\
   \Rightarrow \dfrac{1}{{\sin x}} = \cos ecx \\
   \Rightarrow \dfrac{1}{{\tan x}} = \cot x \\
   \Rightarrow \dfrac{1}{{\cot x}} = \tan x \\
\end{gathered} $

Complete step by step solution:
First simplifying equation using basic trigonometric formula we get,
$ \Rightarrow \dfrac{1}{{\cot x}} = \tan x$
Now, as you know $\tan x = \dfrac{{\sin x}}{{\cos x}}$
After substituting value of $\tan x$ we get,
$ \Rightarrow \dfrac{1}{{\cot x}} = \dfrac{{\sin x}}{{\cos x}}$
And now, you know that $\dfrac{1}{{\cos x}} = \sec x$.
So after substituting the value of $\dfrac{1}{{\cos x}}$into the equation we get,
$ \Rightarrow \dfrac{1}{{\cot x}} = \sin x\sec x$

So here, you can write $\dfrac{1}{{\cot x}}$ as $\sin x\sec x$.

Note:
For these types of questions there is no specific answer. It is possible sometime you solve these questions and get a different answer. This happens because trigonometric functions can be related with each other in many different ways. so always try to simplify these types of questions upto two or three conversions.