Trigonometry Examples

$\frac{1 + \tan (x)}{\sin (x) + \cos (x)}$

Step 1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{1 + \frac{\sin (x)}{\cos (x)}}{\sin (x) + \cos (x)}$

Step 2

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

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Step 2.1

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

$\frac{\cos (x)}{\cos (x)} \cdot \frac{1 + \frac{\sin (x)}{\cos (x)}}{\sin (x) + \cos (x)}$

Step 2.2

Combine.

$\frac{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}{\cos (x) (\sin (x) + \cos (x))}$

Step 3

Apply the distributive property.

$\frac{\cos (x) \cdot 1 + \cos (x) \frac{\sin (x)}{\cos (x)}}{\cos (x) \sin (x) + \cos (x) \cos (x)}$

Step 4

Simplify terms.

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Step 4.1

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

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Step 4.1.1

Cancel the common factor.

$\frac{\cos (x) \cdot 1 + \cos (x) \frac{\sin (x)}{\cos (x)}}{\cos (x) \sin (x) + \cos (x) \cos (x)}$

Step 4.1.2

Rewrite the expression.

$\frac{\cos (x) \cdot 1 + \sin (x)}{\cos (x) \sin (x) + \cos (x) \cos (x)}$

Step 4.2

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

$\frac{\cos (x) + \sin (x)}{\cos (x) \sin (x) + \cos (x) \cos (x)}$

Step 4.3

Factor $\cos (x)$ out of $\cos (x) \sin (x) + \cos (x) \cos (x)$ .

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Step 4.3.1

Factor $\cos (x)$ out of $\cos (x) \sin (x)$ .

$\frac{\cos (x) + \sin (x)}{\cos (x) (\sin (x)) + \cos (x) \cos (x)}$

Step 4.3.2

Factor $\cos (x)$ out of $\cos (x) \cos (x)$ .

$\frac{\cos (x) + \sin (x)}{\cos (x) (\sin (x)) + \cos (x) (\cos (x))}$

Step 4.3.3

Factor $\cos (x)$ out of $\cos (x) (\sin (x)) + \cos (x) (\cos (x))$ .

$\frac{\cos (x) + \sin (x)}{\cos (x) (\sin (x) + \cos (x))}$