Trigonometry Examples

$\frac{\cos (t) - 1}{\sec (t) - 1}$

Step 1

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

$\frac{\cos (t) - 1}{\frac{1}{\cos (t)} - 1}$

Step 2

Multiply the numerator and denominator of the fraction by $\cos (t)$ .

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Step 2.1

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

$\frac{\cos (t)}{\cos (t)} \cdot \frac{\cos (t) - 1}{\frac{1}{\cos (t)} - 1}$

Step 2.2

Combine.

$\frac{\cos (t) (\cos (t) - 1)}{\cos (t) (\frac{1}{\cos (t)} - 1)}$

Step 3

Apply the distributive property.

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

Step 4

Cancel the common factor of $\cos (t)$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $\cos (t) \cos (t)$ .

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Step 5.1.1

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

$\frac{\cos^{1} (t) \cos (t) + \cos (t) \cdot - 1}{1 + \cos (t) \cdot - 1}$

Step 5.1.2

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

$\frac{\cos^{1} (t) \cos^{1} (t) + \cos (t) \cdot - 1}{1 + \cos (t) \cdot - 1}$

Step 5.1.3

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

$\frac{{\cos (t)}^{1 + 1} + \cos (t) \cdot - 1}{1 + \cos (t) \cdot - 1}$

Step 5.1.4

Add $1$ and $1$ .

$\frac{\cos^{2} (t) + \cos (t) \cdot - 1}{1 + \cos (t) \cdot - 1}$

Step 5.2

Move $- 1$ to the left of $\cos (t)$ .

$\frac{\cos^{2} (t) - 1 \cdot \cos (t)}{1 + \cos (t) \cdot - 1}$

Step 5.3

Rewrite $- 1 \cos (t)$ as $- \cos (t)$ .

$\frac{\cos^{2} (t) - \cos (t)}{1 + \cos (t) \cdot - 1}$

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Factor $\cos (t)$ out of $- \cos (t)$ .

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

Step 5.4.3

Factor $\cos (t)$ out of $\cos (t) \cos (t) + \cos (t) \cdot - 1$ .

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

Step 6

Simplify the denominator.

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Step 6.1

Move $- 1$ to the left of $\cos (t)$ .

$\frac{\cos (t) (\cos (t) - 1)}{1 - 1 \cdot \cos (t)}$

Step 6.2

Rewrite $- 1 \cos (t)$ as $- \cos (t)$ .

$\frac{\cos (t) (\cos (t) - 1)}{1 - \cos (t)}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $\cos (t) - 1$ and $1 - \cos (t)$ .

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Step 7.1.1

Factor $- 1$ out of $\cos (t)$ .

$\frac{\cos (t) (- 1 (- \cos (t)) - 1)}{1 - \cos (t)}$

Step 7.1.2

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{\cos (t) (- 1 (- \cos (t)) - 1 (1))}{1 - \cos (t)}$

Step 7.1.3

Factor $- 1$ out of $- 1 (- \cos (t)) - 1 (1)$ .

$\frac{\cos (t) (- 1 (- \cos (t) + 1))}{1 - \cos (t)}$

Step 7.1.4

Reorder terms.

$\frac{\cos (t) (- 1 (1 - \cos (t)))}{1 - \cos (t)}$

Step 7.1.5

Cancel the common factor.

$\frac{\cos (t) (- 1 (1 - \cos (t)))}{1 - \cos (t)}$

Step 7.1.6

Divide $\cos (t) \cdot (- 1)$ by $1$ .

$\cos (t) \cdot (- 1)$

Step 7.2

Simplify the expression.

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Step 7.2.1

Move $- 1$ to the left of $\cos (t)$ .

$- 1 \cdot \cos (t)$

Step 7.2.2

Rewrite $- 1 \cos (t)$ as $- \cos (t)$ .

$- \cos (t)$