Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Add $1$ and $1$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

Add $1$ and $1$ .

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

Add $\cos^{2} (x)$ and $0$ .

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

Step 5.8

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

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Step 5.8.1

Rearrange terms.

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

Step 5.8.2

Apply pythagorean identity.

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

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

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

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