Question

Simplify $\dfrac{{1 + \cos 2x}}{{\sin 2x}}$

Answer 1

Hint: The simplify must be done using the correct formula. The numerator and the denominator must be possibly simplified using formulae. The formula for the $\cos 2x$ is simplified into the multiple of $2{\cos ^2}x$ and $1$ . The formula for the $\sin 2x$ is simplified into multiple $2$ , $\sin x$and $\cos x$ .
The simplification must contain one only term.

Formula used:
$\cos 2x = 2{\cos ^2}x - 1$
$\sin 2x = 2\sin x\cos x$

Complete step-by-step answer:
Given,
The term which is needed to simplify is $\dfrac{{1 + \cos 2x}}{{\sin 2x}}$
Let us assume the term which is needed to be $I = \dfrac{{1 + \cos 2x}}{{\sin 2x}}$
Substitute $\cos 2x = 2{\cos ^2}x - 1$ in the above equation
$I = \dfrac{{1 + 2{{\cos }^2}x - 1}}{{\sin 2x}}$
Substitute $\sin 2x = 2\sin x\cos x$ in the above equation
$I = \dfrac{{1 + 2{{\cos }^2}x - 1}}{{2\sin x\cos x}}$
Subtract the terms in numerator from the above equation, we get
$I = \dfrac{{2{{\cos }^2}x}}{{2\sin x\cos x}}$
The value in the numerator and the denominator in the above equation, we get
$I = \dfrac{{{{\cos }^2}x}}{{\sin x\cos x}}$
The values in the numerator must be split into multiplication of two terms,
$I = \dfrac{{\cos x \times \cos x}}{{\sin x\cos x}}$
The value in the numerator and denominator is divided in the above equation,
$I = \dfrac{{\cos x}}{{\sin x}}$
$\cot x$is the ratio of $\cos x$and $\sin x$ , i.e $\cot x$ is the ratio of adjacent angle to opposite angle.
$\cot x$ is the reciprocal of $\tan x$ .
$I = \cot x$
The term which needs to be simplified into a single term.
$\dfrac{{1 + \cos 2x}}{{\sin 2x}} = \cot x$

Note: The formulae should be known well. The simplification is done in a single term. The formula for the $\cos 2x$ is simplified into the multiple of $2{\cos ^2}x$ and $1$ . The formula for the $\sin 2x$ is simplified into multiple $2$ , $\sin x$ and $\cos x$ . The simplification must contain one only term. $\cot x$ is the ratio of adjacent angle to opposite angle. $\cot x$ is the reciprocal of $\tan x$ . $\cot x$ is the ratio of opposite angle to adjacent angle.