Simplify (sin(u)*cos(u)^2+sin(u)^3)/(sin(u)^2)

$\frac{\sin (u) \cdot \cos^{2} (u) + \sin^{3} (u)}{\sin^{2} (u)}$

Step 1

Simplify the numerator.

Factor $\sin (u)$ out of $\sin (u) \cdot \cos^{2} (u) + \sin^{3} (u)$ .

Factor $\sin (u)$ out of $\sin (u) \cdot \cos^{2} (u)$ .

$\frac{\sin (u) (\cos^{2} (u)) + \sin^{3} (u)}{\sin^{2} (u)}$

Factor $\sin (u)$ out of $\sin^{3} (u)$ .

$\frac{\sin (u) (\cos^{2} (u)) + \sin (u) \sin^{2} (u)}{\sin^{2} (u)}$

Factor $\sin (u)$ out of $\sin (u) (\cos^{2} (u)) + \sin (u) \sin^{2} (u)$ .

$\frac{\sin (u) (\cos^{2} (u) + \sin^{2} (u))}{\sin^{2} (u)}$

Rearrange terms.

$\frac{\sin (u) (\sin^{2} (u) + \cos^{2} (u))}{\sin^{2} (u)}$

Apply pythagorean identity.

$\frac{\sin (u) \cdot 1}{\sin^{2} (u)}$

Step 2

Reduce the expression by cancelling the common factors.

Multiply $\sin (u)$ by $1$ .

$\frac{\sin (u)}{\sin^{2} (u)}$

Cancel the common factor of $\sin (u)$ and $\sin^{2} (u)$ .

Multiply by $1$ .

$\frac{\sin (u) \cdot 1}{\sin^{2} (u)}$

Cancel the common factors.

Factor $\sin (u)$ out of $\sin^{2} (u)$ .

$\frac{\sin (u) \cdot 1}{\sin (u) \sin (u)}$

Cancel the common factor.

$\frac{\sin (u) \cdot 1}{\sin (u) \sin (u)}$

Rewrite the expression.

$\frac{1}{\sin (u)}$

Step 3

Convert from $\frac{1}{\sin (u)}$ to $\csc (u)$ .

$\csc (u)$