Trigonometry Examples

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

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

Step 1

Simplify each term.

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Step 1.1

Rewrite

sec (x)

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

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

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

Step 1.2

Rewrite

tan (x)

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

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

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

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

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

Step 2

Expand

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

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

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

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))

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

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))

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

\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))

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

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)}

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

1

$1$ .

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

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

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))

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

Step 3.1.3

Combine

sin (x)

$\sin (x)$ and

\frac{1}{cos (x)}

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

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

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

Step 3.1.4

Multiply

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

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

1

$1$ .

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

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

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)

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

Step 3.1.6

Multiply

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

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

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Step 3.1.6.1

Combine

sin (x)

$\sin (x)$ and

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

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

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

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

Step 3.1.6.2

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

Step 3.1.6.3

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

Step 3.1.6.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 3.1.6.5

Add

1

$1$ and

1

$1$ .

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

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

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

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

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

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

Step 3.2

Add

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

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

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

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

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

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

Step 3.3

Add

\frac{1}{cos (x)}

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

0

$0$ .

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

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

Step 5

Apply pythagorean identity.

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

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

Step 6

Cancel the common factor of

{cos}^{2} (x)

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

cos (x)

$\cos (x)$ .

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Step 6.1

Factor

cos (x)

$\cos (x)$ out of

{cos}^{2} (x)

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

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

$1$ .

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

$1$ .

$\cos (x)$