Question

How do you simplify \[\dfrac{{\cot x}}{{\csc x}}\]?

Answer 1

Hint: Here we will simplify the expression by using trigonometric identities and mathematical operations. First, we will rewrite the numerator and denominator in terms of sine and cosine functions by using the reciprocal and quotient identities. We will then simplify the equation to get the required answer.

Complete step-by-step answer:
We have to simplify \[\dfrac{{\cot x}}{{\csc x}}\].
We will first convert both the functions into sine and cosine functions.
We know that, \[\cot x = \dfrac{{\cos x}}{{\sin x}}\] and \[\csc x = \dfrac{1}{{\sin x}}\].
Now we will substitute above value in the expression \[\dfrac{{\cot x}}{{\csc x}}\]. Therefore, we get
\[\dfrac{{\cot x}}{{\csc x}} = \dfrac{{\dfrac{{\cos x}}{{\sin x}}}}{{\dfrac{1}{{\sin x}}}}\]
Rewriting the above equation, we get
\[ \Rightarrow \dfrac{{\cot x}}{{\csc x}} = \dfrac{{\cos x}}{{\sin x}} \times \dfrac{{\sin x}}{1}\]
Cancelling out the common terms, we get
\[ \Rightarrow \dfrac{{\cot x}}{{\csc x}} = \cos x\]

Therefore, the value of \[\dfrac{{\cot x}}{{\csc x}}\] is \[\cos x\].

Additional information:
Trigonometry is a branch of mathematics that deals with specific functions of angles and also their application in calculations and simplification. There are six important trigonometry functions and they are sine, cosine, tangent, cotangent, secant, and cosecant. Identities are those equations that are true for every variable. Trigonometry is used in practical life to construct buildings, it is used in the field of engineering, finding height and distances, etc.

Note:
We can also solve this question using an alternate method.
We will first square the given expression \[\dfrac{{\cot x}}{{\csc x}}\].
\[{\left( {\dfrac{{\cot x}}{{\csc x}}} \right)^2} = \dfrac{{{{\cot }^2}x}}{{{{\csc }^2}x}}\]
We know that \[{\cot ^2}x = {\csc ^2}x - 1\].
Substituting this trigonometric identity in the above equation, we get
\[ \Rightarrow {\left( {\dfrac{{\cot x}}{{\csc x}}} \right)^2} = \dfrac{{{{\csc }^2}x - 1}}{{{{\csc }^2}x}}\]
Rewriting the equation, we get
\[ \Rightarrow {\left( {\dfrac{{\cot x}}{{\csc x}}} \right)^2} = \dfrac{{{{\csc }^2}x}}{{{{\csc }^2}x}} - \dfrac{1}{{{{\csc }^2}x}}\]
Dividing the terms and using the reciprocal function \[\csc x = \dfrac{1}{{\sin x}}\], we get
\[ \Rightarrow {\left( {\dfrac{{\cot x}}{{\csc x}}} \right)^2} = 1 - {\sin ^2}x\]
Now using the trigonometric identity \[{\cos ^2}x = 1 - {\sin ^2}x\] in the above equation, we get
\[ \Rightarrow {\left( {\dfrac{{\cot x}}{{\csc x}}} \right)^2} = {\cos ^2}x\]
Taking square root on both sides, we get
\[ \Rightarrow \left( {\dfrac{{\cot x}}{{\csc x}}} \right) = \cos x\]
Therefore, the value of \[\dfrac{{\cot x}}{{\csc x}}\] is \[\cos x\].