Trigonometry Examples

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

Step 1

Subtract $\frac{\sin (x)}{\sin (x) + \cos (x)}$ from both sides of the equation.

$\frac{\tan (x)}{1 + \tan (x)} - \frac{\sin (x)}{\sin (x) + \cos (x)} = 0$

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$\frac{\frac{\sin (x)}{\cos (x)}}{1 + \tan (x)} - \frac{\sin (x)}{\sin (x) + \cos (x)} = 0$

Step 2.1.2

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

$\frac{\frac{\sin (x)}{\cos (x)}}{1 + \frac{\sin (x)}{\cos (x)}} - \frac{\sin (x)}{\sin (x) + \cos (x)} = 0$

Step 2.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{1 + \frac{\sin (x)}{\cos (x)}} - \frac{\sin (x)}{\sin (x) + \cos (x)} = 0$

Step 2.1.4

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

$\frac{\sin (x)}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})} - \frac{\sin (x)}{\sin (x) + \cos (x)} = 0$

Step 2.2

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

$\frac{\sin (x)}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})} \cdot \frac{\sin (x) + \cos (x)}{\sin (x) + \cos (x)} - \frac{\sin (x)}{\sin (x) + \cos (x)} = 0$

Step 2.3

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

$\frac{\sin (x)}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})} \cdot \frac{\sin (x) + \cos (x)}{\sin (x) + \cos (x)} - \frac{\sin (x)}{\sin (x) + \cos (x)} \cdot \frac{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})} = 0$

Step 2.4

Write each expression with a common denominator of $\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

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

$\frac{\sin (x) (\sin (x) + \cos (x))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} - \frac{\sin (x)}{\sin (x) + \cos (x)} \cdot \frac{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})} = 0$

Step 2.4.2

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

$\frac{\sin (x) (\sin (x) + \cos (x))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} - \frac{\sin (x) (\cos (x) (1 + \frac{\sin (x)}{\cos (x)}))}{(\sin (x) + \cos (x)) (\cos (x) (1 + \frac{\sin (x)}{\cos (x)}))} = 0$

Step 2.4.3

Reorder the factors of $(\sin (x) + \cos (x)) (\cos (x) (1 + \frac{\sin (x)}{\cos (x)}))$ .

$\frac{\sin (x) (\sin (x) + \cos (x))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} - \frac{\sin (x) (\cos (x) (1 + \frac{\sin (x)}{\cos (x)}))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{\sin (x) (\sin (x) + \cos (x)) - \sin (x) (\cos (x) (1 + \frac{\sin (x)}{\cos (x)}))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Factor $\sin (x)$ out of $\sin (x) (\sin (x) + \cos (x)) - \sin (x) \cos (x) (1 + \frac{\sin (x)}{\cos (x)})$ .

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Step 2.6.1.1

Factor $\sin (x)$ out of $- \sin (x) \cos (x) (1 + \frac{\sin (x)}{\cos (x)})$ .

$\frac{\sin (x) (\sin (x) + \cos (x)) + \sin (x) (- \cos (x) (1 + \frac{\sin (x)}{\cos (x)}))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.6.1.2

Factor $\sin (x)$ out of $\sin (x) (\sin (x) + \cos (x)) + \sin (x) (- \cos (x) (1 + \frac{\sin (x)}{\cos (x)}))$ .

$\frac{\sin (x) (\sin (x) + \cos (x) - \cos (x) (1 + \frac{\sin (x)}{\cos (x)}))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.6.2

Apply the distributive property.

$\frac{\sin (x) (\sin (x) + \cos (x) - \cos (x) \cdot 1 - \cos (x) \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.6.3

Multiply $- 1$ by $1$ .

$\frac{\sin (x) (\sin (x) + \cos (x) - \cos (x) - \cos (x) \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.6.4

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

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Step 2.6.4.1

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

$\frac{\sin (x) (\sin (x) + \cos (x) - \cos (x) + \cos (x) \cdot - 1 \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.6.4.2

Cancel the common factor.

$\frac{\sin (x) (\sin (x) + \cos (x) - \cos (x) + \cos (x) \cdot - 1 \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.6.4.3

Rewrite the expression.

$\frac{\sin (x) (\sin (x) + \cos (x) - \cos (x) - \sin (x))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.6.5

Subtract $\sin (x)$ from $\sin (x)$ .

$\frac{\sin (x) (0 + \cos (x) - \cos (x))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.6.6

Add $0$ and $\cos (x)$ .

$\frac{\sin (x) (\cos (x) - \cos (x))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.6.7

Subtract $\cos (x)$ from $\cos (x)$ .

$\frac{\sin (x) \cdot 0}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.7

Multiply $\sin (x)$ by $0$ .

$\frac{0}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Step 2.8

Divide $0$ by $\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))$ .

$0 = 0$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.