To simplify the expression:
[tex]\[
\sin x + \cos x \cot x
\][/tex]
we start by recalling the definition of the cotangent function in terms of sine and cosine:
[tex]\[
\cot x = \frac{\cos x}{\sin x}
\][/tex]
Next, we substitute [tex]\(\cot x\)[/tex] with [tex]\(\frac{\cos x}{\sin x}\)[/tex] in the original expression:
[tex]\[
\sin x + \cos x \cot x = \sin x + \cos x \left(\frac{\cos x}{\sin x}\right)
\][/tex]
Simplifying inside the parentheses, we get:
[tex]\[
\sin x + \cos x \left(\frac{\cos x}{\sin x}\right) = \sin x + \frac{\cos^2 x}{\sin x}
\][/tex]
To combine these terms, we need a common denominator. The common denominator is [tex]\(\sin x\)[/tex], so we rewrite the first term with this common denominator:
[tex]\[
\sin x = \frac{\sin^2 x}{\sin x}
\][/tex]
Now our expression looks like this:
[tex]\[
\frac{\sin^2 x}{\sin x} + \frac{\cos^2 x}{\sin x}
\][/tex]
Since the denominators are the same, we can combine the numerators over this common denominator:
[tex]\[
\frac{\sin^2 x + \cos^2 x}{\sin x}
\][/tex]
We use the Pythagorean identity, which states that:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
Therefore, our expression simplifies to:
[tex]\[
\frac{1}{\sin x}
\][/tex]
Thus, we recognize that [tex]\(\frac{1}{\sin x}\)[/tex] is the definition of the cosecant function:
[tex]\[
\csc x
\][/tex]
So the simplified expression is:
[tex]\[
\sin x + \cos x \cot x = \csc x
\][/tex]
