Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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

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

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Rewrite using the commutative property of multiplication.

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

Step 3.1.6

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

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Step 3.1.6.1

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

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

Step 3.1.6.2

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

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

Step 3.1.6.3

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

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

Step 3.1.6.4

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

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

Step 3.1.6.5

Add $1$ and $1$ .

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

Step 3.2

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

$\frac{1}{\cos (x)} + 0 - \frac{\sin^{2} (x)}{\cos (x)}$

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Apply pythagorean identity.

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

Step 6

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

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Step 6.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Multiply by $1$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 6.2.4

Divide $\cos (x)$ by $1$ .

$\cos (x)$