Simplify sin(A)+(cos(A)^2)/(1+sin(A))

$\sin (A) + \frac{\cos^{2} (A)}{1 + \sin (A)}$

Step 1

To write $\sin (A)$ as a fraction with a common denominator, multiply by $\frac{1 + \sin (A)}{1 + \sin (A)}$ .

$\frac{\sin (A) (1 + \sin (A))}{1 + \sin (A)} + \frac{\cos^{2} (A)}{1 + \sin (A)}$

Step 2

Combine the numerators over the common denominator.

$\frac{\sin (A) (1 + \sin (A)) + \cos^{2} (A)}{1 + \sin (A)}$

Step 3

Simplify the numerator.

Apply the distributive property.

$\frac{\sin (A) \cdot 1 + \sin (A) \sin (A) + \cos^{2} (A)}{1 + \sin (A)}$

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

$\frac{\sin (A) + \sin (A) \sin (A) + \cos^{2} (A)}{1 + \sin (A)}$

Multiply $\sin (A) \sin (A)$ .

Raise $\sin (A)$ to the power of $1$ .

$\frac{\sin (A) + \sin^{1} (A) \sin (A) + \cos^{2} (A)}{1 + \sin (A)}$

Raise $\sin (A)$ to the power of $1$ .

$\frac{\sin (A) + \sin^{1} (A) \sin^{1} (A) + \cos^{2} (A)}{1 + \sin (A)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sin (A) + {\sin (A)}^{1 + 1} + \cos^{2} (A)}{1 + \sin (A)}$

Add $1$ and $1$ .

$\frac{\sin (A) + \sin^{2} (A) + \cos^{2} (A)}{1 + \sin (A)}$

Apply pythagorean identity.

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

Step 4

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

$1$