Simplify (1+csc(t))/(1+sin(t))

$\frac{1 + \csc (t)}{1 + \sin (t)}$

Rewrite $\csc (t)$ in terms of sines and cosines.

$\frac{1 + \frac{1}{\sin (t)}}{1 + \sin (t)}$

Multiply the numerator and denominator of the complex fraction by $\sin (t)$ .

Multiply $\frac{1 + \frac{1}{\sin (t)}}{1 + \sin (t)}$ by $\frac{\sin (t)}{\sin (t)}$ .

$\frac{\sin (t)}{\sin (t)} \cdot \frac{1 + \frac{1}{\sin (t)}}{1 + \sin (t)}$

Combine.

$\frac{\sin (t) (1 + \frac{1}{\sin (t)})}{\sin (t) (1 + \sin (t))}$

Apply the distributive property.

$\frac{\sin (t) \cdot 1 + \sin (t) \frac{1}{\sin (t)}}{\sin (t) \cdot 1 + \sin (t) \sin (t)}$

Simplify terms.

Cancel the common factor of $\sin (t)$ .

Cancel the common factor.

$\frac{\sin (t) \cdot 1 + \sin (t) \frac{1}{\sin (t)}}{\sin (t) \cdot 1 + \sin (t) \sin (t)}$

Rewrite the expression.

$\frac{\sin (t) \cdot 1 + 1}{\sin (t) \cdot 1 + \sin (t) \sin (t)}$

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

$\frac{\sin (t) + 1}{\sin (t) \cdot 1 + \sin (t) \sin (t)}$

Factor $\sin (t)$ out of $\sin (t) \cdot 1 + \sin (t) \sin (t)$ .

Factor $\sin (t)$ out of $\sin (t) \cdot 1$ .

$\frac{\sin (t) + 1}{\sin (t) (1) + \sin (t) \sin (t)}$

Factor $\sin (t)$ out of $\sin (t) \sin (t)$ .

$\frac{\sin (t) + 1}{\sin (t) (1) + \sin (t) (\sin (t))}$

Factor $\sin (t)$ out of $\sin (t) (1) + \sin (t) (\sin (t))$ .

$\frac{\sin (t) + 1}{\sin (t) (1 + \sin (t))}$

Cancel the common factor of $\sin (t) + 1$ and $1 + \sin (t)$ .

Reorder terms.

$\frac{1 + \sin (t)}{\sin (t) (1 + \sin (t))}$

Cancel the common factor.

$\frac{1 + \sin (t)}{\sin (t) (1 + \sin (t))}$

Rewrite the expression.

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

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

$\csc (t)$