Trigonometry Examples

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

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

Step 1

Simplify the numerator.

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Step 1.1

Combine

\frac{1}{y}

$\frac{1}{y}$ and

sin (x)

$\sin (x)$ .

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

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

Step 1.2

Combine

sin (y)

$\sin (y)$ and

\frac{1}{y}

$\frac{1}{y}$ .

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

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

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

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

Step 2

Simplify the denominator.

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Step 2.1

Combine

\frac{1}{y}

$\frac{1}{y}$ and

cos (x)

$\cos (x)$ .

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

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

Step 2.2

Combine

sin (y)

$\sin (y)$ and

\frac{x}{y}

$\frac{x}{y}$ .

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

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

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

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

Step 3

Multiply the numerator and denominator of the fraction by

y

$y$ .

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Step 3.1

Multiply

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

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

\frac{y}{y}

$\frac{y}{y}$ .

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

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

Step 3.2

Combine.

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

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

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

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

Step 4

Apply the distributive property.

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

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

Step 5

Simplify by cancelling.

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Step 5.1

Cancel the common factor of

y

$y$ .

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Step 5.1.1

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

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

Step 5.2

Cancel the common factor of

$y$ .

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Step 5.2.1

Move the leading negative in

$- \frac{\sin (y)}{y}$ into the numerator.

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 5.3

Cancel the common factor of

$y$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

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

Step 5.4

Cancel the common factor of

$y$ .

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Step 5.4.1

Move the leading negative in

$- \frac{\sin (y) x}{y}$ into the numerator.

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

Step 5.4.2

Cancel the common factor.

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

Step 5.4.3

Rewrite the expression.

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

Step 6

Reorder factors in

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

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