Precalculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify terms.

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Step 4.1

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

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

Reorder terms.

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Divide $\sin (y)$ by $1$ .

$\sin (y)$