What is the value of `(tan x)/((sin^3 x)/(cos x) + sin x cos x )`

Answer

To solve the expression \(\frac{\tan x}{\frac{\sin^3 x}{\cos x} + \sin x \cos x}\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the given expression:
\[ \frac{\tan x}{\frac{\sin^3 x}{\cos x} + \sin x \cos x} \]
Recall that \(\tan x = \frac{\sin x}{\cos x}\). Substituting this into the expression gives:
\[ \frac{\frac{\sin x}{\cos x}}{\frac{\sin^3 x}{\cos x} + \sin x \cos x} \]

### Step 2: Simplify the denominator
Now, we simplify the denominator:
\[ \frac{\sin^3 x}{\cos x} + \sin x \cos x \]
To combine these terms, we can factor out \(\sin x\):
\[ = \sin x \left( \frac{\sin^2 x}{\cos x} + \cos x \right) \]
Now, we can rewrite \(\cos x\) as \(\frac{\cos^2 x}{\cos x}\):
\[ = \sin x \left( \frac{\sin^2 x + \cos^2 x}{\cos x} \right) \]
Using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), we have:
\[ = \sin x \left( \frac{1}{\cos x} \right) = \frac{\sin x}{\cos x} \]

### Step 3: Substitute back into the expression
Now we substitute the simplified denominator back into the expression:
\[ \frac{\frac{\sin x}{\cos x}}{\frac{\sin x}{\cos x}} \]

### Step 4: Simplify the expression
This simplifies to:
\[ 1 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{1} \]