To simplify the expression [tex]\(\frac{\csc x}{\cot x \sec x}\)[/tex], we can use trigonometric identities. Let's break it down step-by-step:
1. Recall the trigonometric identities:
[tex]\[
\csc x = \frac{1}{\sin x}
\][/tex]
[tex]\[
\cot x = \frac{\cos x}{\sin x}
\][/tex]
[tex]\[
\sec x = \frac{1}{\cos x}
\][/tex]
2. Substitute these identities into the given expression:
[tex]\[
\frac{\csc x}{\cot x \sec x} = \frac{\frac{1}{\sin x}}{\left(\frac{\cos x}{\sin x}\right) \left(\frac{1}{\cos x}\right)}
\][/tex]
3. Simplify the denominator first:
[tex]\[
\left(\frac{\cos x}{\sin x}\right) \left(\frac{1}{\cos x}\right) = \frac{\cos x}{\sin x} \cdot \frac{1}{\cos x}
\][/tex]
[tex]\[
= \frac{\cos x \cdot 1}{\sin x \cdot \cos x} = \frac{1}{\sin x}
\][/tex]
4. Now the expression is:
[tex]\[
\frac{\frac{1}{\sin x}}{\frac{1}{\sin x}}
\][/tex]
Which can be rewritten as:
[tex]\[
\frac{1/\sin x}{1/\sin x} = 1
\][/tex]
Therefore, the simplified expression is:
[tex]\[
1
\][/tex]
