Question

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

Answer 1

Hint : First of all we will write the expansion of $\tan x$ as we know that $\tan x$ can be expressed in terms of $\sin x$ and $\cos x$ hence $\tan x=\dfrac{\sin x}{\cos x}$ by this we can write the inverse of $\tan x$ , then we can find the value of $\dfrac{1}{\tan x}$ by using the trigonometric identities and we can obtain the given result.

Complete step-by-step answer :
Trigonometric functions are also known as the circular functions.The basic trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant. The angles of sine, cosine, and tangent are the primary classification of functions of trigonometry. And the three functions which are cotangent, secant and cosecant can be derived from the primary functions.
Trigonometry is a discipline of mathematics that investigates the relationship between triangle side lengths and angles. Applications of geometry to astronomical research gave rise to the field in the Hellenistic civilization during the third century BC.
The Greeks concentrated on chord calculations, whereas Indian mathematicians developed the first documented tables of values for trigonometric ratios like sine.
Now according to the question:
As we know in mathematics $\tan x=\dfrac{perpendicular}{base}$
And we can also write $\tan x=\dfrac{\sin x}{\cos x}$
Hence we will simplify $\dfrac{1}{\tan x}$ as:
$\Rightarrow \dfrac{1}{\tan x}=\dfrac{1}{\left( \dfrac{\sin x}{\cos x} \right)}$
$\Rightarrow \dfrac{1}{\tan x}=\dfrac{\cos x}{\sin x}$
And we know that we can write $\dfrac{\cos x}{\sin x}=\cot x$ hence:
$\Rightarrow \dfrac{1}{\tan x}=\cot x$
Hence in simplified way we can write $\dfrac{1}{\tan x}=\cot x$

Note : We must keep one thing in mind that $\sin \theta$ is not the same as $\sin \times \theta$ because it represents a ratio, not a product and this is true for all the trigonometric ratios. Any trigonometric function of agle ${{\theta }^{\circ }}$ is equal to the same trigonometric function of any angle $n\times {{360}^{\circ }}+\theta$, where $n$ is any integer.