Trigonometry Examples

$\frac{\sin (x) + \sin (y)}{\cos (x) - \cos (y)} = \cot (x)$

Step 1

Subtract $\cot (x)$ from both sides of the equation.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Write each expression with a common denominator of $(\cos (x) - \cos (y)) \sin (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

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

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

Step 5.2

Multiply $\frac{\cos (x)}{\sin (x)}$ by $\frac{\cos (x) - \cos (y)}{\cos (x) - \cos (y)}$ .

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

Step 5.3

Reorder the factors of $(\cos (x) - \cos (y)) \sin (x)$ .

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Apply the distributive property.

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

Step 7.2

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

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Step 7.2.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 7.2.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 7.2.3

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

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

Step 7.2.4

Add $1$ and $1$ .

$\frac{\sin^{2} (x) + \sin (y) \sin (x) - \cos (x) (\cos (x) - \cos (y))}{\sin (x) (\cos (x) - \cos (y))} = 0$

Step 7.3

Apply the distributive property.

$\frac{\sin^{2} (x) + \sin (y) \sin (x) - \cos (x) \cos (x) - \cos (x) (- \cos (y))}{\sin (x) (\cos (x) - \cos (y))} = 0$

Step 7.4

Multiply $- \cos (x) \cos (x)$ .

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Step 7.4.1

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

$\frac{\sin^{2} (x) + \sin (y) \sin (x) - (\cos^{1} (x) \cos (x)) - \cos (x) (- \cos (y))}{\sin (x) (\cos (x) - \cos (y))} = 0$

Step 7.4.2

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

$\frac{\sin^{2} (x) + \sin (y) \sin (x) - (\cos^{1} (x) \cos^{1} (x)) - \cos (x) (- \cos (y))}{\sin (x) (\cos (x) - \cos (y))} = 0$

Step 7.4.3

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

$\frac{\sin^{2} (x) + \sin (y) \sin (x) - {\cos (x)}^{1 + 1} - \cos (x) (- \cos (y))}{\sin (x) (\cos (x) - \cos (y))} = 0$

Step 7.4.4

Add $1$ and $1$ .

$\frac{\sin^{2} (x) + \sin (y) \sin (x) - \cos^{2} (x) - \cos (x) (- \cos (y))}{\sin (x) (\cos (x) - \cos (y))} = 0$

Step 7.5

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

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Step 7.5.1

Multiply $- 1$ by $- 1$ .

$\frac{\sin^{2} (x) + \sin (y) \sin (x) - \cos^{2} (x) + 1 \cos (x) \cos (y)}{\sin (x) (\cos (x) - \cos (y))} = 0$

Step 7.5.2

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

$\frac{\sin^{2} (x) + \sin (y) \sin (x) - \cos^{2} (x) + \cos (x) \cos (y)}{\sin (x) (\cos (x) - \cos (y))} = 0$