Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $\sin (t) \sin (t)$ .

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

Add $1$ and $1$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $- 1$ by $1$ .

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

Step 6.4

Multiply $- - \cos (t)$ .

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Step 6.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.4.2

Multiply $\cos (t)$ by $1$ .

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

Step 6.5

Reorder $\sin^{2} (t)$ and $- 1$ .

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

Step 6.6

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

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

Step 6.7

Factor $- 1$ out of $\sin^{2} (t)$ .

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

Step 6.8

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (t))$ .

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

Step 6.9

Rewrite $- 1 (1 - \sin^{2} (t))$ as $- (1 - \sin^{2} (t))$ .

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

Step 6.10

Apply pythagorean identity.

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

Step 6.11

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

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Step 6.11.1

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

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

Step 6.11.2

Multiply by $1$ .

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

Step 6.11.3

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

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

Step 7

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

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Step 7.1

Reorder terms.

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

$\frac{\cos (t)}{\sin (t)}$

Step 8

Convert from $\frac{\cos (t)}{\sin (t)}$ to $\cot (t)$ .

$\cot (t)$