Calculus Examples

$\frac{\cot (\frac{7 π}{4}) - \cot (\frac{5 π}{4})}{\frac{7 π}{4} - \frac{5 π}{4}}$

Step 1

Multiply the numerator and denominator of the fraction by $4$ .

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Step 1.1

Multiply $\frac{\cot (\frac{7 π}{4}) - \cot (\frac{5 π}{4})}{\frac{7 π}{4} - \frac{5 π}{4}}$ by $\frac{4}{4}$ .

$\frac{4}{4} \cdot \frac{\cot (\frac{7 π}{4}) - \cot (\frac{5 π}{4})}{\frac{7 π}{4} - \frac{5 π}{4}}$

Step 1.2

Combine.

$\frac{4 (\cot (\frac{7 π}{4}) - \cot (\frac{5 π}{4}))}{4 (\frac{7 π}{4} - \frac{5 π}{4})}$

Step 2

Apply the distributive property.

$\frac{4 \cot (\frac{7 π}{4}) + 4 (- \cot (\frac{5 π}{4}))}{4 \frac{7 π}{4} + 4 (- \frac{5 π}{4})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $4$ .

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Step 3.1.1

Cancel the common factor.

$\frac{4 \cot (\frac{7 π}{4}) + 4 (- \cot (\frac{5 π}{4}))}{4 \frac{7 π}{4} + 4 (- \frac{5 π}{4})}$

Step 3.1.2

Rewrite the expression.

$\frac{4 \cot (\frac{7 π}{4}) + 4 (- \cot (\frac{5 π}{4}))}{7 π + 4 (- \frac{5 π}{4})}$

Step 3.2

Cancel the common factor of $4$ .

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Step 3.2.1

Move the leading negative in $- \frac{5 π}{4}$ into the numerator.

$\frac{4 \cot (\frac{7 π}{4}) + 4 (- \cot (\frac{5 π}{4}))}{7 π + 4 \frac{- 5 π}{4}}$

Step 3.2.2

Cancel the common factor.

$\frac{4 \cot (\frac{7 π}{4}) + 4 (- \cot (\frac{5 π}{4}))}{7 π + 4 \frac{- 5 π}{4}}$

Step 3.2.3

Rewrite the expression.

$\frac{4 \cot (\frac{7 π}{4}) + 4 (- \cot (\frac{5 π}{4}))}{7 π - 5 π}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cotangent is negative in the fourth quadrant.

$\frac{4 (- \cot (\frac{π}{4})) + 4 (- \cot (\frac{5 π}{4}))}{7 π - 5 π}$

Step 4.2

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$\frac{4 (- 1 \cdot 1) + 4 (- \cot (\frac{5 π}{4}))}{7 π - 5 π}$

Step 4.3

Multiply $4 (- 1 \cdot 1)$ .

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Step 4.3.1

Multiply $- 1$ by $1$ .

$\frac{4 \cdot - 1 + 4 (- \cot (\frac{5 π}{4}))}{7 π - 5 π}$

Step 4.3.2

Multiply $4$ by $- 1$ .

$\frac{- 4 + 4 (- \cot (\frac{5 π}{4}))}{7 π - 5 π}$

Step 4.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{- 4 + 4 (- \cot (\frac{π}{4}))}{7 π - 5 π}$

Step 4.5

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$\frac{- 4 + 4 (- 1 \cdot 1)}{7 π - 5 π}$

Step 4.6

Multiply $4 (- 1 \cdot 1)$ .

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Step 4.6.1

Multiply $- 1$ by $1$ .

$\frac{- 4 + 4 \cdot - 1}{7 π - 5 π}$

Step 4.6.2

Multiply $4$ by $- 1$ .

$\frac{- 4 - 4}{7 π - 5 π}$

Step 4.7

Subtract $4$ from $- 4$ .

$\frac{- 8}{7 π - 5 π}$

Step 5

Simplify terms.

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Step 5.1

Subtract $5 π$ from $7 π$ .

$\frac{- 8}{2 π}$

Step 5.2

Cancel the common factor of $- 8$ and $2$ .

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Step 5.2.1

Factor $2$ out of $- 8$ .

$\frac{2 \cdot - 4}{2 π}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2 π$ .

$\frac{2 \cdot - 4}{2 (π)}$

Step 5.2.2.2

Cancel the common factor.

$\frac{2 \cdot - 4}{2 π}$

Step 5.2.2.3

Rewrite the expression.

$\frac{- 4}{π}$

Step 5.3

Move the negative in front of the fraction.

$- \frac{4}{π}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{4}{π}$

Decimal Form:

$- 1.27323954 \dots$