Precalculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Raise $\cos (y)$ to the power of $1$ .

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

Step 5.1.2

Raise $\cos (y)$ to the power of $1$ .

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

Step 5.1.3

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

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

Step 5.1.4

Add $1$ and $1$ .

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

Factor $\cos (y)$ out of $\cos^{2} (y)$ .

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

Step 5.3.2

Multiply by $1$ .

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

Step 5.3.3

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

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

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

$\cos (y)$