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How do you use the fundamental identities to simplify \[\dfrac{{cotx}}{{cscx}}\] ?

Last updated date: 25th Feb 2024
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IVSAT 2024
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Hint:In order to solve the question we need to understand the mathematical statement of the Question especially the term fundamental identities . The fundamental trigonometric identities are the basic identities which are taken to establish other relationships among trigonometric functions .
There are six basic trigonometric ratios that are as follows - sine, cosine, tangent, cosecant, secant and cotangent . These six trigonometric ratios are abbreviated as sin, cos, tan, csc, sec, cot . And there are some valid relationships between these ratios .

Complete Step by Step Solution :
So , first we are going to simplify the numerator cot $x$.
cot $x$can be expressed in the form of having other relationships with tan$x$ratio . As we know the cotangent is reciprocal of tangent . The tangent can be expressed in the form of other two trigonometric ratios sine , cosine and both the relationships can be expressed as follows –
cot $x$ =$\dfrac{1}{{\tan x}}$ also tan $x$= $\dfrac{{\sin x}}{{\cos x}}$
Now rewriting the cot $x$ in the relationship of two trigonometric ratios sine and cosine .
cot $x$ = $\dfrac{{\cos x}}{{\sin x}}$.
Also Coming to the denominator , we can write cosec $x$in the form of sine as it is the reciprocal of sine , which can be expressed as = csc $x$= $\dfrac{1}{{\sin x}}$
Therefore , now combining both the numerator and denominator \[\dfrac{{cotx}}{{cscx}}\] which we expressed in the form of basic trigonometric ratio as sine and cosine , so that we can simplify the given ratio as
= $\dfrac{{\dfrac{{\cos x}}{{\sin x}}}}{{\dfrac{1}{{\sin x}}}}$
= $\dfrac{{\cos x}}{{\sin x \bullet \dfrac{1}{{\sin x}}}}$
= cos $x$
Therefore , the Simplification of the \[\dfrac{{cotx}}{{cscx}}\] is cos $x$.

Note : Always remember that whenever it is asked to simplify, convert the given ratio into the basic form of trigonometric ratio that is sine and cosine .
Then To Divide this and get the answer into its simplest form , take the first fraction as it is and flip or reciprocal the second fraction that is numerator will become denominator and denominator will become numerator .
Applying these steps for solving the question