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

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

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

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

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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Apply the distributive property.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Apply the distributive property.

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

Apply the distributive property.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

Subtract $\sin^{2} (x)$ from $0$ .

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

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

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

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

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

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

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

Apply pythagorean identity.

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

Multiply $- 1$ by $1$ .

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

Step 6

Separate fractions.

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

Step 7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{- 1}{\sin (x)} \sec (x)$

Step 8

Separate fractions.

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

Step 9

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{- 1}{1} \csc (x) \sec (x)$

Step 10

Divide $- 1$ by $1$ .

$- \csc (x) \sec (x)$