Sure, let's simplify the given expression step-by-step!
The expression given is:
[tex]\[
\frac{\frac{\sin x}{\cos x}}{\sin^2 x}
\][/tex]
First, let's rewrite the expression to make it easier to simplify:
[tex]\[
\frac{\sin x / \cos x}{\sin^2 x} = \frac{\sin x}{\cos x \cdot \sin^2 x}
\][/tex]
Next, let's see how we can simplify the denominator:
[tex]\[
\cos x \cdot \sin^2 x = \cos x \cdot (\sin x \cdot \sin x) = \cos x \sin x^2
\][/tex]
So now our expression looks like this:
[tex]\[
\frac{\sin x}{\cos x \sin^2 x}
\][/tex]
Divide both the numerator and the denominator by [tex]\(\sin x\)[/tex]:
[tex]\[
\frac{1}{\cos x \sin x}
\][/tex]
Recognize that using the double-angle identity for sine, [tex]\(\sin(2x) = 2\sin x \cos x\)[/tex], we have:
[tex]\[
\sin x \cos x = \frac{\sin(2x)}{2}
\][/tex]
So:
[tex]\[
\cos x \sin x = \frac{\sin(2x)}{2}
\][/tex]
Thus, our expression now becomes:
[tex]\[
\frac{1}{(\frac{\sin(2x)}{2})}
\][/tex]
Invert the denominator and simplify:
[tex]\[
\frac{1}{\frac{\sin(2x)}{2}} = \frac{2}{\sin(2x)}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{2}{\sin(2x)}
\][/tex]
So the final answer is:
[tex]\[
\frac{2}{\sin(2x)}
\][/tex]
