Trigonometry Examples

$(\sin (x) - \cos (x)) (\sec (x) + \csc (x))$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 1.2

Rewrite $\csc (x)$ in terms of sines and cosines.

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

Step 2

Expand $(\sin (x) - \cos (x)) (\frac{1}{\cos (x)} + \frac{1}{\sin (x)})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine $\sin (x)$ and $\frac{1}{\cos (x)}$ .

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

Step 3.1.2

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

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

Rewrite the expression.

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

Step 3.1.3

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

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Step 3.1.3.1

Factor $\cos (x)$ out of $- \cos (x)$ .

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

Step 3.1.3.2

Cancel the common factor.

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

Step 3.1.3.3

Rewrite the expression.

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

Step 3.1.4

Combine $\frac{1}{\sin (x)}$ and $\cos (x)$ .

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

Step 3.2

Subtract $1$ from $1$ .

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

Step 3.3

Add $\frac{\sin (x)}{\cos (x)}$ and $0$ .

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

Step 4

Simplify each term.

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Step 4.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 4.2

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\tan (x) - \cot (x)$