Precalculus Examples

$\csc (u) - \cot (u) = \frac{\sin (u)}{1 + \cos (u)}$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\frac{1}{\sin (u)} - \cot (u) = \frac{\sin (u)}{1 + \cos (u)}$

Step 1.1.2

Rewrite $\cot (u)$ in terms of sines and cosines.

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

Step 2

Multiply both sides of the equation by $\sin (u)$ .

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

Step 3

Apply the distributive property.

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

Step 4

Cancel the common factor of $\sin (u)$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

Cancel the common factor of $\sin (u)$ .

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Step 6.1

Factor $\sin (u)$ out of $- \sin (u)$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Multiply $\sin (u) \frac{\sin (u)}{1 + \cos (u)}$ .

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 8

Subtract $\frac{\sin^{2} (u)}{1 + \cos (u)}$ from both sides of the equation.

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

Step 9

Simplify $1 - \cos (u) - \frac{\sin^{2} (u)}{1 + \cos (u)}$ .

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Step 9.1

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

$1 - \cos (u) \cdot \frac{1 + \cos (u)}{1 + \cos (u)} - \frac{\sin^{2} (u)}{1 + \cos (u)} = 0$

Step 9.2

Combine $- \cos (u)$ and $\frac{1 + \cos (u)}{1 + \cos (u)}$ .

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

Step 9.3

Combine the numerators over the common denominator.

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

Step 9.4

Simplify each term.

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Step 9.4.1

Simplify the numerator.

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Step 9.4.1.1

Apply the distributive property.

$1 + \frac{- \cos (u) \cdot 1 - \cos (u) \cos (u) - \sin^{2} (u)}{1 + \cos (u)} = 0$

Step 9.4.1.2

Multiply $- 1$ by $1$ .

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

Step 9.4.1.3

Multiply $- \cos (u) \cos (u)$ .

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Step 9.4.1.3.1

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

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

Step 9.4.1.3.2

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

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

Step 9.4.1.3.3

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

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

Step 9.4.1.3.4

Add $1$ and $1$ .

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

Step 9.4.1.4

Factor $- 1$ out of $- \cos^{2} (u)$ .

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

Step 9.4.1.5

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

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

Step 9.4.1.6

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

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

Step 9.4.1.7

Rearrange terms.

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

Step 9.4.1.8

Apply pythagorean identity.

$1 + \frac{- \cos (u) - 1 \cdot 1}{1 + \cos (u)} = 0$

Step 9.4.1.9

Multiply $- 1$ by $1$ .

$1 + \frac{- \cos (u) - 1}{1 + \cos (u)} = 0$

Step 9.4.1.10

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

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Step 9.4.1.10.1

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

$1 + \frac{- (\cos (u)) - 1}{1 + \cos (u)} = 0$

Step 9.4.1.10.2

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

$1 + \frac{- (\cos (u)) - 1 (1)}{1 + \cos (u)} = 0$

Step 9.4.1.10.3

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

$1 + \frac{- (\cos (u) + 1)}{1 + \cos (u)} = 0$

Step 9.4.2

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

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Step 9.4.2.1

Reorder terms.

$1 + \frac{- (1 + \cos (u))}{1 + \cos (u)} = 0$

Step 9.4.2.2

Cancel the common factor.

$1 + \frac{- (1 + \cos (u))}{1 + \cos (u)} = 0$

Step 9.4.2.3

Divide $- 1$ by $1$ .

$1 - 1 = 0$

Step 9.5

Subtract $1$ from $1$ .

$0 = 0$

Step 10

Since $0 = 0$ , the equation will always be true for any value of $u$ .

All real numbers

Step 11

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$