Trigonometry Examples

$(\sec (x) - \tan (x)) (\csc (x) + 1) = \cot (x)$

Step 1

Start on the left side.

$(\sec (x) - \tan (x)) (\csc (x) + 1)$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Apply the distributive property.

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

Step 2.3.3

Apply the distributive property.

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

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\sin (x)}$ .

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

Step 2.4.1.2

Multiply $\frac{1}{\cos (x)}$ by $1$ .

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

Step 2.4.1.3

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

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Step 2.4.1.3.1

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

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

Step 2.4.1.3.2

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

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

Step 2.4.1.3.3

Cancel the common factor.

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

Step 2.4.1.3.4

Rewrite the expression.

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

Step 2.4.1.4

Move the negative in front of the fraction.

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

Step 2.4.1.5

Multiply $- 1$ by $1$ .

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

Step 2.4.2

Subtract $\frac{1}{\cos (x)}$ from $\frac{1}{\cos (x)}$ .

$\frac{1}{\cos (x) \sin (x)} + 0 - \frac{\sin (x)}{\cos (x)}$

Step 2.4.3

Add $\frac{1}{\cos (x) \sin (x)}$ and $0$ .

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

Step 3

Subtract fractions.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 4

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

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

Step 5

Apply pythagorean identity.

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

Step 6

Cancel the common factor of ${\cos (x)}^{2}$ and $\cos (x)$ .

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

Step 7

Rewrite $\frac{\cos (x)}{\sin (x)}$ as $\cot (x)$ .

$\cot (x)$

Step 8

Because the two sides have been shown to be equivalent, the equation is an identity.

$(\sec (x) - \tan (x)) (\csc (x) + 1) = \cot (x)$ is an identity