Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- \sin (θ) \sin (θ)$ .

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Add $1$ and $1$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Multiply $- \cos (θ) \cos (θ)$ .

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.5.4

Add $1$ and $1$ .

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

Subtract $\sin^{2} (θ)$ from $0$ .

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

Step 5.9

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

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

Step 5.10

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

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

Step 5.11

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

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

Step 5.12

Apply pythagorean identity.

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

Step 5.13

Multiply $- 1$ by $1$ .

$\frac{- 1}{\cos (θ) \sin (θ)}$

Step 6

Separate fractions.

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

Step 7

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

$\frac{- 1}{\sin (θ)} \sec (θ)$

Step 8

Separate fractions.

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

Step 9

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

$\frac{- 1}{1} \csc (θ) \sec (θ)$

Step 10

Divide $- 1$ by $1$ .

$- \csc (θ) \sec (θ)$