Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Add $1$ and $1$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

Add $1$ and $1$ .

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

Step 5.5

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

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

Step 5.6

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

$\frac{2 \cos (x) \sin (x) - \sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x)}$

Step 5.7

Rewrite $2 \cos (x) \sin (x) - \sin^{2} (x) - \cos^{2} (x)$ in a factored form.

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Step 5.7.1

Reorder $2 \cos (x)$ and $\sin (x)$ .

$\frac{\sin (x) (2 \cos (x)) - \sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x)}$

Step 5.7.2

Reorder $\sin (x)$ and $2$ .

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

Step 5.7.3

Apply the sine double-angle identity.

$\frac{\sin (2 x) - \sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x)}$

Step 5.8

Factor $- 1$ out of $- \sin^{2} (x)$ .

$\frac{\sin (2 x) - (\sin^{2} (x)) - \cos^{2} (x)}{\cos (x) \sin (x)}$

Step 5.9

Factor $- 1$ out of $- \cos^{2} (x)$ .

$\frac{\sin (2 x) - (\sin^{2} (x)) - (\cos^{2} (x))}{\cos (x) \sin (x)}$

Step 5.10

Factor $- 1$ out of $- (\sin^{2} (x)) - (\cos^{2} (x))$ .

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

Step 5.11

Apply pythagorean identity.

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

Step 5.12

Multiply $- 1$ by $1$ .

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