Trigonometry Examples

$\frac{\sin^{2} (x) - 4}{2 \cos (x) + 1} \cdot \frac{10 \cos (x) + 5}{2 \sin (x) + 4}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $4$ as $2^{2}$ .

$\frac{\sin^{2} (x) - 2^{2}}{2 \cos (x) + 1} \cdot \frac{10 \cos (x) + 5}{2 \sin (x) + 4}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (x)$ and $b = 2$ .

$\frac{(\sin (x) + 2) (\sin (x) - 2)}{2 \cos (x) + 1} \cdot \frac{10 \cos (x) + 5}{2 \sin (x) + 4}$

Step 2

Simplify terms.

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Step 2.1

Factor $5$ out of $10 \cos (x) + 5$ .

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Step 2.1.1

Factor $5$ out of $10 \cos (x)$ .

$\frac{(\sin (x) + 2) (\sin (x) - 2)}{2 \cos (x) + 1} \cdot \frac{5 (2 \cos (x)) + 5}{2 \sin (x) + 4}$

Step 2.1.2

Factor $5$ out of $5$ .

$\frac{(\sin (x) + 2) (\sin (x) - 2)}{2 \cos (x) + 1} \cdot \frac{5 (2 \cos (x)) + 5 (1)}{2 \sin (x) + 4}$

Step 2.1.3

Factor $5$ out of $5 (2 \cos (x)) + 5 (1)$ .

$\frac{(\sin (x) + 2) (\sin (x) - 2)}{2 \cos (x) + 1} \cdot \frac{5 (2 \cos (x) + 1)}{2 \sin (x) + 4}$

Step 2.2

Simplify terms.

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Step 2.2.1

Factor $2$ out of $2 \sin (x) + 4$ .

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Step 2.2.1.1

Factor $2$ out of $2 \sin (x)$ .

$\frac{(\sin (x) + 2) (\sin (x) - 2)}{2 \cos (x) + 1} \cdot \frac{5 (2 \cos (x) + 1)}{2 (\sin (x)) + 4}$

Step 2.2.1.2

Factor $2$ out of $4$ .

$\frac{(\sin (x) + 2) (\sin (x) - 2)}{2 \cos (x) + 1} \cdot \frac{5 (2 \cos (x) + 1)}{2 \sin (x) + 2 \cdot 2}$

Step 2.2.1.3

Factor $2$ out of $2 \sin (x) + 2 \cdot 2$ .

$\frac{(\sin (x) + 2) (\sin (x) - 2)}{2 \cos (x) + 1} \cdot \frac{5 (2 \cos (x) + 1)}{2 (\sin (x) + 2)}$

Step 2.2.2

Cancel the common factor of $\sin (x) + 2$ .

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Step 2.2.2.1

Factor $\sin (x) + 2$ out of $2 (\sin (x) + 2)$ .

$\frac{(\sin (x) + 2) (\sin (x) - 2)}{2 \cos (x) + 1} \cdot \frac{5 (2 \cos (x) + 1)}{(\sin (x) + 2) \cdot 2}$

Step 2.2.2.2

Cancel the common factor.

$\frac{(\sin (x) + 2) (\sin (x) - 2)}{2 \cos (x) + 1} \cdot \frac{5 (2 \cos (x) + 1)}{(\sin (x) + 2) \cdot 2}$

Step 2.2.2.3

Rewrite the expression.

$\frac{\sin (x) - 2}{2 \cos (x) + 1} \cdot \frac{5 (2 \cos (x) + 1)}{2}$

Step 2.2.3

Cancel the common factor of $2 \cos (x) + 1$ .

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Step 2.2.3.1

Factor $2 \cos (x) + 1$ out of $5 (2 \cos (x) + 1)$ .

$\frac{\sin (x) - 2}{2 \cos (x) + 1} \cdot \frac{(2 \cos (x) + 1) \cdot 5}{2}$

Step 2.2.3.2

Cancel the common factor.

$\frac{\sin (x) - 2}{2 \cos (x) + 1} \cdot \frac{(2 \cos (x) + 1) \cdot 5}{2}$

Step 2.2.3.3

Rewrite the expression.

$(\sin (x) - 2) \cdot \frac{5}{2}$

Step 2.2.4

Apply the distributive property.

$\sin (x) \frac{5}{2} - 2 (\frac{5}{2})$

Step 2.2.5

Combine $\sin (x)$ and $\frac{5}{2}$ .

$\frac{\sin (x) \cdot 5}{2} - 2 (\frac{5}{2})$

Step 2.2.6

Cancel the common factor of $2$ .

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Step 2.2.6.1

Factor $2$ out of $- 2$ .

$\frac{\sin (x) \cdot 5}{2} + 2 (- 1) \frac{5}{2}$

Step 2.2.6.2

Cancel the common factor.

$\frac{\sin (x) \cdot 5}{2} + 2 \cdot - 1 \frac{5}{2}$

Step 2.2.6.3

Rewrite the expression.

$\frac{\sin (x) \cdot 5}{2} - 1 \cdot 5$

Step 2.2.7

Multiply $- 1$ by $5$ .

$\frac{\sin (x) \cdot 5}{2} - 5$

Step 2.3

Move $5$ to the left of $\sin (x)$ .

$\frac{5 \sin (x)}{2} - 5$