Question

How do you simplify $\cot x+\tan x$ ?

Answer 1

Here we have been asked to simplify the given trigonometric expression $\cot x+\tan x$ . For doing that we will use the following valid trigonometric formulae $\tan x=\dfrac{\sin x}{\cos x}$ , $\cot x=\dfrac{\cos x}{\sin x}$, ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ and $\sin 2x=2\sin x\cos x$ .

Complete step by step answer:
Now considering from the question we have been asked to simplify the given trigonometric expression $\cot x+\tan x$ .
For doing that we will use the following valid trigonometric formulae $\tan x=\dfrac{\sin x}{\cos x}$ , $\cot x=\dfrac{\cos x}{\sin x}$, ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ and $\sin 2x=2\sin x\cos x$ which we have learnt during the basic concepts.
Now by substituting trigonometric formulae $\tan x=\dfrac{\sin x}{\cos x}$ and $\cot x=\dfrac{\cos x}{\sin x}$ in the given expression $\cot x+\tan x$ we will have $\Rightarrow \dfrac{\cos x}{\sin x}+\dfrac{\sin x}{\cos x}=\dfrac{{{\sin }^{2}}x+{{\cos }^{2}}x}{\sin x\cos x}$ .
By using ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ this expression can be further simplified as $\dfrac{1}{\sin x\cos x}$ .
Now by using $\sin 2x=2\sin x\cos x$ we will have $\dfrac{2}{\sin 2x}$ .
From the basic concepts of trigonometry we know that the cosecant function is the reciprocal of sine function which can be mathematically expressed as $\csc x=\dfrac{1}{\sin x}$ .
Now by using $\csc x=\dfrac{1}{\sin x}$ we will have $\Rightarrow 2\csc 2x$

Therefore we can conclude that the simplified and reduced form of given trigonometric expression $\cot x+\tan x$ is $2\csc 2x$

Note: While answering questions of this type we should be sure with the trigonometric concepts that we are going to apply in the process. This is a very simple and easy question and can be answered accurately in a short span of time. Very few mistakes are possible in questions of this type. Someone can forget some of the simplification formula to apply in between and end up having an incomplete answer for example if we had not used the formula ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ then we will unable to completely simplify the given expression.