Question

How do you simplify $\dfrac{{1 + \csc x}}{{\cos x + \cot x}}$?

Answer 1

Hint:
This question is related to the trigonometry, and we have to simplify the expression and this can be done by using trigonometric identities, such as , $\cot x = \dfrac{{\cos x}}{{\sin x}}$ and $\csc x = \dfrac{1}{{\sin x}}$, and then further simplification of the expression we will get the required result.

Complete step by step solution:
Given $\dfrac{{1 + \csc x}}{{\cos x + \cot x}}$,
We know that $\cot x = \dfrac{{\cos x}}{{\sin x}}$, and $\csc x = \dfrac{1}{{\sin x}}$, and by substituting the identities in the given expression, we get,
$ \Rightarrow \dfrac{{1 + \dfrac{1}{{\sin x}}}}{{\cos x + \dfrac{{\cos x}}{{\sin x}}}}$,
Now taking Least Common Multiple in both the numerator and denominator we get,
$ \Rightarrow \dfrac{{\dfrac{{1 + \sin x}}{{\sin x}}}}{{\dfrac{{\sin \cos x + \cos x}}{{\sin x}}}}$,
Now eliminating the like terms we get,
$ \Rightarrow \dfrac{{1 + \sin x}}{{\sin x\cos x + \cos x}}$,
Now taking out the common term in the denominator we get,
$ \Rightarrow \dfrac{{1 + \sin x}}{{\cos x\left( {1 + \sin x} \right)}}$,
Now again eliminating the like terms we get,
$ \Rightarrow \dfrac{1}{{\cos x}}$,
And by using the trigonometric identity i.e.,
$ \Rightarrow \dfrac{1}{{\cos x}} = \sec x$,
From the above we can say that $\dfrac{{1 + \csc x}}{{\cos x + \cot x}} = \sec x$,
So, the simplified form will be $\sec x$.

$\therefore $The simplified form of the given expression $\dfrac{{1 + \csc x}}{{\cos x + \cot x}}$ will be equal to $\sec x$.

Note:
An identity is an equation that always holds true. A trigonometric identity is an identity that contains trigonometric functions and holds true for all right-angled triangles. They are useful when solving questions with trigonometric functions and expressions. There are six trigonometric ratios, sine, cosine, tangent, cosecant, secant and cotangent. These six trigonometric ratios are abbreviated as sin, cos, tan, csc, sec, cot. These are referred to as ratios since they can be expressed in terms of the sides of a right-angled triangle for a specific angle θ. There are many trigonometric identities, here are some useful identities:
${\sin ^2}x = 1 - {\cos ^2}x$,
${\cos ^2}x + {\sin ^2}x = 1$,
${\sec ^2}x - {\tan ^2}x = 1$,
${\csc ^2}x = 1 + {\cot ^2}x$,
${\cos ^2}x - {\sin ^2}x = 1 - 2{\sin ^2}x$,
${\cos ^2}x - {\sin ^2}x = 2{\cos ^2}x - 1$,
$\sin 2x = 2\sin x\cos x$,
$2{\cos ^2}x = 1 + \cos 2x$,
$\tan 2x = \dfrac{{2\tan x}}{{1 - {{\tan }^2}x}}$.