Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

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

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

Step 2.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.3.1.3

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

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

Step 2.3.1.4

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

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

Step 2.3.1.5

Rewrite using the commutative property of multiplication.

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

Step 2.3.1.6

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

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Step 2.3.1.6.1

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

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

Step 2.3.1.6.2

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

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

Step 2.3.1.6.3

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

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

Step 2.3.1.6.4

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

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

Step 2.3.1.6.5

Add $1$ and $1$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Apply pythagorean identity.

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Multiply by $1$ .

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

Step 2.6.2.2

Cancel the common factor.

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

Step 2.6.2.3

Rewrite the expression.

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

Step 2.6.2.4

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

$\cos (B)$

Step 3

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

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