Precalculus Examples

\frac{1 + \cos (y)}{1 + \sec (y)}

$\frac{1 + \cos (y)}{1 + \sec (y)}$

Step 1

Rewrite

\sec (y)

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

\frac{1 + \cos (y)}{1 + \frac{1}{\cos (y)}}

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

Step 2

Multiply the numerator and denominator of the fraction by

\cos (y)

$\cos (y)$ .

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Step 2.1

Multiply

\frac{1 + \cos (y)}{1 + \frac{1}{\cos (y)}}

$\frac{1 + \cos (y)}{1 + \frac{1}{\cos (y)}}$ by

\frac{\cos (y)}{\cos (y)}

$\frac{\cos (y)}{\cos (y)}$ .

\frac{\cos (y)}{\cos (y)} \cdot \frac{1 + \cos (y)}{1 + \frac{1}{\cos (y)}}

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

Step 2.2

Combine.

\frac{\cos (y) (1 + \cos (y))}{\cos (y) (1 + \frac{1}{\cos (y)})}

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

\frac{\cos (y) (1 + \cos (y))}{\cos (y) (1 + \frac{1}{\cos (y)})}

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

Step 3

Apply the distributive property.

\frac{\cos (y) \cdot 1 + \cos (y) \cos (y)}{\cos (y) \cdot 1 + \cos (y) \frac{1}{\cos (y)}}

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of

\cos (y)

$\cos (y)$ .

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Step 4.1.1

Cancel the common factor.

\frac{\cos (y) \cdot 1 + \cos (y) \cos (y)}{\cos (y) \cdot 1 + \cos (y) \frac{1}{\cos (y)}}

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

Step 4.1.2

Rewrite the expression.

\frac{\cos (y) \cdot 1 + \cos (y) \cos (y)}{\cos (y) \cdot 1 + 1}

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

\frac{\cos (y) \cdot 1 + \cos (y) \cos (y)}{\cos (y) \cdot 1 + 1}

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

Step 4.2

Factor

\cos (y)

$\cos (y)$ out of

\cos (y) \cdot 1 + \cos (y) \cos (y)

$\cos (y) \cdot 1 + \cos (y) \cos (y)$ .

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Step 4.2.1

Factor

\cos (y)

$\cos (y)$ out of

\cos (y) \cdot 1

$\cos (y) \cdot 1$ .

\frac{\cos (y) (1) + \cos (y) \cos (y)}{\cos (y) \cdot 1 + 1}

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

Step 4.2.2

Factor

\cos (y)

$\cos (y)$ out of

\cos (y) \cos (y)

$\cos (y) \cos (y)$ .

\frac{\cos (y) (1) + \cos (y) (\cos (y))}{\cos (y) \cdot 1 + 1}

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

Step 4.2.3

Factor

\cos (y)

$\cos (y)$ out of

\cos (y) (1) + \cos (y) (\cos (y))

$\cos (y) (1) + \cos (y) (\cos (y))$ .

\frac{\cos (y) (1 + \cos (y))}{\cos (y) \cdot 1 + 1}

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

\frac{\cos (y) (1 + \cos (y))}{\cos (y) \cdot 1 + 1}

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

Step 4.3

Multiply

\cos (y)

$\cos (y)$ by

1

$1$ .

\frac{\cos (y) (1 + \cos (y))}{\cos (y) + 1}

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

Step 4.4

Cancel the common factor of

1 + \cos (y)

$1 + \cos (y)$ and

\cos (y) + 1

$\cos (y) + 1$ .

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Step 4.4.1

Reorder terms.

\frac{\cos (y) (1 + \cos (y))}{1 + \cos (y)}

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

Step 4.4.2

Cancel the common factor.

\frac{\cos (y) (1 + \cos (y))}{1 + \cos (y)}

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

Step 4.4.3

Divide

\cos (y)

$\cos (y)$ by

1

$1$ .

\cos (y)

$\cos (y)$

\cos (y)

$\cos (y)$

\cos (y)

$\cos (y)$