Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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

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

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Rewrite using the commutative property of multiplication.

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

Step 3.1.6

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

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Step 3.1.6.1

Move the leading negative in $- \frac{1}{\sin (x)}$ into the numerator.

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

Step 3.1.6.2

Cancel the common factor.

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

Step 3.1.6.3

Rewrite the expression.

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

Step 3.2

Subtract $1$ from $1$ .

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

Step 3.3

Subtract $\frac{\sin (x)}{\cos (x)}$ from $0$ .

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

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 $\cot (x)$ .

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