Trigonometry Examples

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

Step 1

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

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

Step 2

To write $\frac{\cos (θ)}{1 - \sin (θ)}$ as a fraction with a common denominator, multiply by $\frac{\cos (θ)}{\cos (θ)}$ .

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

Step 3

To write $- \frac{\sin (θ)}{\cos (θ)}$ as a fraction with a common denominator, multiply by $\frac{1 - \sin (θ)}{1 - \sin (θ)}$ .

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

Step 4

Write each expression with a common denominator of $(1 - \sin (θ)) \cos (θ)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(1 - \sin (θ)) \cos (θ)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $\cos (θ) \cos (θ)$ .

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Step 6.1.1

Raise $\cos (θ)$ to the power of $1$ .

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

Step 6.1.2

Raise $\cos (θ)$ to the power of $1$ .

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

Step 6.1.3

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

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

Step 6.1.4

Add $1$ and $1$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $- 1$ by $1$ .

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

Step 6.4

Multiply $- \sin (θ) (- \sin (θ))$ .

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Step 6.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.4.2

Multiply $\sin (θ)$ by $1$ .

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

Step 6.4.3

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

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

Step 6.4.4

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

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

Step 6.4.5

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

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

Step 6.4.6

Add $1$ and $1$ .

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

Step 6.5

Rewrite $\cos^{2} (θ) - \sin (θ) + \sin^{2} (θ)$ in a factored form.

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Step 6.5.1

Rearrange terms.

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

Step 6.5.2

Rearrange terms.

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

Step 6.5.3

Apply pythagorean identity.

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

Step 7

Cancel the common factor of $- \sin (θ) + 1$ and $1 - \sin (θ)$ .

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Step 7.1

Reorder terms.

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

$\frac{1}{\cos (θ)}$

Step 8

Convert from $\frac{1}{\cos (θ)}$ to $\sec (θ)$ .

$\sec (θ)$