Trigonometry Examples

(\sec (B) + \tan (B)) (1 - \sin (B)) = \cos (B)

$(\sec (B) + \tan (B)) (1 - \sin (B)) = \cos (B)$

Step 1

Start on the left side.

(\sec (B) + \tan (B)) (1 - \sin (B))

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

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 (B)

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

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

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

Step 2.1.2

Rewrite

\tan (B)

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

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

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

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

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

Step 2.2

Expand

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

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

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Step 2.2.1

Apply the distributive property.

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

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

Step 2.2.2

Apply the distributive property.

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

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

Step 2.2.3

Apply the distributive property.

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

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

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

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

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply

\frac{1}{\cos (B)}

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

1

$1$ .

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

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

Step 2.3.1.2

Rewrite using the commutative property of multiplication.

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

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

Step 2.3.1.3

Combine

\sin (B)

$\sin (B)$ and

\frac{1}{\cos (B)}

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

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

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

Step 2.3.1.4

Multiply

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

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

1

$1$ .

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

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

Step 2.3.1.5

Rewrite using the commutative property of multiplication.

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

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

Step 2.3.1.6

Multiply

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

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

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Step 2.3.1.6.1

Combine

\sin (B)

$\sin (B)$ and

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

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

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

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

Step 2.3.1.6.2

Raise

\sin (B)

$\sin (B)$ to the power of

1

$1$ .

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

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

Step 2.3.1.6.3

Raise

\sin (B)

$\sin (B)$ to the power of

1

$1$ .

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

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

Step 2.3.1.6.4

Use the power rule

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

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

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

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

Step 2.3.1.6.5

Add

1

$1$ and

1

$1$ .

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

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

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

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

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

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

Step 2.3.2

Add

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

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

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

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

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

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

Step 2.3.3

Add

\frac{1}{\cos (B)}

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

0

$0$ .

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

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

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

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

Step 2.4

Combine the numerators over the common denominator.

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

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

Step 2.5

Apply pythagorean identity.

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

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

Step 2.6

Cancel the common factor of

\cos^{2} (B)

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

\cos (B)

$\cos (B)$ .

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Step 2.6.1

Factor

\cos (B)

$\cos (B)$ out of

\cos^{2} (B)

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

\frac{\cos (B) \cos (B)}{\cos (B)}

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

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Multiply by

1

$1$ .

\frac{\cos (B) \cos (B)}{\cos (B) \cdot 1}

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

Step 2.6.2.2

Cancel the common factor.

\frac{\cos (B) \cos (B)}{\cos (B) \cdot 1}

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

Step 2.6.2.3

Rewrite the expression.

\frac{\cos (B)}{1}

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

Step 2.6.2.4

Divide

\cos (B)

$\cos (B)$ by

1

$1$ .

\cos (B)

$\cos (B)$

\cos (B)

$\cos (B)$

\cos (B)

$\cos (B)$

\cos (B)

$\cos (B)$

Step 3

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

(\sec (B) + \tan (B)) (1 - \sin (B)) = \cos (B)

$(\sec (B) + \tan (B)) (1 - \sin (B)) = \cos (B)$ is an identity