Calculus Examples

$\frac{1 + \tan (x)}{1 + \cot (x)}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1 + \tan (x)$ and $g (x) = 1 + \cot (x)$ .

$\frac{(1 + \cot (x)) \frac{d}{d x} [1 + \tan (x)] - (1 + \tan (x)) \frac{d}{d x} [1 + \cot (x)]}{{(1 + \cot (x))}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $1 + \tan (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\tan (x)]$ .

$\frac{(1 + \cot (x)) (\frac{d}{d x} [1] + \frac{d}{d x} [\tan (x)]) - (1 + \tan (x)) \frac{d}{d x} [1 + \cot (x)]}{{(1 + \cot (x))}^{2}}$

Step 2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{(1 + \cot (x)) (0 + \frac{d}{d x} [\tan (x)]) - (1 + \tan (x)) \frac{d}{d x} [1 + \cot (x)]}{{(1 + \cot (x))}^{2}}$

Step 2.3

Add $0$ and $\frac{d}{d x} [\tan (x)]$ .

$\frac{(1 + \cot (x)) \frac{d}{d x} [\tan (x)] - (1 + \tan (x)) \frac{d}{d x} [1 + \cot (x)]}{{(1 + \cot (x))}^{2}}$

Step 3

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\frac{(1 + \cot (x)) \sec^{2} (x) - (1 + \tan (x)) \frac{d}{d x} [1 + \cot (x)]}{{(1 + \cot (x))}^{2}}$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $1 + \cot (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\cot (x)]$ .

$\frac{(1 + \cot (x)) \sec^{2} (x) - (1 + \tan (x)) (\frac{d}{d x} [1] + \frac{d}{d x} [\cot (x)])}{{(1 + \cot (x))}^{2}}$

Step 4.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{(1 + \cot (x)) \sec^{2} (x) - (1 + \tan (x)) (0 + \frac{d}{d x} [\cot (x)])}{{(1 + \cot (x))}^{2}}$

Step 4.3

Add $0$ and $\frac{d}{d x} [\cot (x)]$ .

$\frac{(1 + \cot (x)) \sec^{2} (x) - (1 + \tan (x)) \frac{d}{d x} [\cot (x)]}{{(1 + \cot (x))}^{2}}$

Step 5

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$\frac{(1 + \cot (x)) \sec^{2} (x) - (1 + \tan (x)) (- \csc^{2} (x))}{{(1 + \cot (x))}^{2}}$

Step 6

Multiply.

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Step 6.1

Multiply $- 1$ by $- 1$ .

$\frac{(1 + \cot (x)) \sec^{2} (x) + 1 (1 + \tan (x)) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 6.2

Multiply $1 + \tan (x)$ by $1$ .

$\frac{(1 + \cot (x)) \sec^{2} (x) + (1 + \tan (x)) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{1 \sec^{2} (x) + \cot (x) \sec^{2} (x) + (1 + \tan (x)) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.2

Apply the distributive property.

$\frac{1 \sec^{2} (x) + \cot (x) \sec^{2} (x) + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Multiply $\sec^{2} (x)$ by $1$ .

$\frac{\sec^{2} (x) + \cot (x) \sec^{2} (x) + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.2

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

$\frac{\sec^{2} (x) + \frac{\cos (x)}{\sin (x)} \sec^{2} (x) + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.3

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

$\frac{\sec^{2} (x) + \frac{\cos (x)}{\sin (x)} {(\frac{1}{\cos (x)})}^{2} + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.4

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{\sec^{2} (x) + \frac{\cos (x)}{\sin (x)} \cdot \frac{1^{2}}{\cos^{2} (x)} + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.5

One to any power is one.

$\frac{\sec^{2} (x) + \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos^{2} (x)} + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.6

Multiply $\frac{\cos (x)}{\sin (x)}$ by $\frac{1}{\cos^{2} (x)}$ .

$\frac{\sec^{2} (x) + \frac{\cos (x)}{\sin (x) \cos^{2} (x)} + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.7

Cancel the common factor of $\cos (x)$ and $\cos^{2} (x)$ .

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Step 7.3.1.7.1

Multiply by $1$ .

$\frac{\sec^{2} (x) + \frac{\cos (x) \cdot 1}{\sin (x) \cos^{2} (x)} + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.7.2

Cancel the common factors.

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Step 7.3.1.7.2.1

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

$\frac{\sec^{2} (x) + \frac{\cos (x) \cdot 1}{\cos (x) (\sin (x) \cos (x))} + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.7.2.2

Cancel the common factor.

$\frac{\sec^{2} (x) + \frac{\cos (x) \cdot 1}{\cos (x) (\sin (x) \cos (x))} + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.7.2.3

Rewrite the expression.

$\frac{\sec^{2} (x) + \frac{1}{\sin (x) \cos (x)} + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.8

Separate fractions.

$\frac{\sec^{2} (x) + \frac{1}{\cos (x)} \cdot \frac{1}{\sin (x)} + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.9

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{\sec^{2} (x) + \frac{1}{\cos (x)} \csc (x) + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.10

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

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + 1 \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.11

Multiply $\csc^{2} (x)$ by $1$ .

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \tan (x) \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.12

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

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \frac{\sin (x)}{\cos (x)} \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.13

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

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \frac{\sin (x)}{\cos (x)} {(\frac{1}{\sin (x)})}^{2}}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.14

Apply the product rule to $\frac{1}{\sin (x)}$ .

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \frac{\sin (x)}{\cos (x)} \cdot \frac{1^{2}}{\sin^{2} (x)}}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.15

One to any power is one.

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\sin^{2} (x)}}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.16

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{1}{\sin^{2} (x)}$ .

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \frac{\sin (x)}{\cos (x) \sin^{2} (x)}}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.17

Cancel the common factor of $\sin (x)$ and $\sin^{2} (x)$ .

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Step 7.3.1.17.1

Multiply by $1$ .

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \frac{\sin (x) \cdot 1}{\cos (x) \sin^{2} (x)}}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.17.2

Cancel the common factors.

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Step 7.3.1.17.2.1

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

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \frac{\sin (x) \cdot 1}{\sin (x) (\cos (x) \sin (x))}}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.17.2.2

Cancel the common factor.

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \frac{\sin (x) \cdot 1}{\sin (x) (\cos (x) \sin (x))}}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.17.2.3

Rewrite the expression.

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \frac{1}{\cos (x) \sin (x)}}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.18

Separate fractions.

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \frac{1}{\sin (x)} \cdot \frac{1}{\cos (x)}}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.19

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

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \frac{1}{\sin (x)} \sec (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.1.20

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{\sec^{2} (x) + \sec (x) \csc (x) + \csc^{2} (x) + \csc (x) \sec (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.2

Reorder the factors of $\sec (x) \csc (x)$ .

$\frac{\sec^{2} (x) + \csc (x) \sec (x) + \csc^{2} (x) + \csc (x) \sec (x)}{{(1 + \cot (x))}^{2}}$

Step 7.3.3

Add $\csc (x) \sec (x)$ and $\csc (x) \sec (x)$ .

$\frac{\sec^{2} (x) + 2 \csc (x) \sec (x) + \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.4

Reorder terms.

$\frac{2 \csc (x) \sec (x) + \sec^{2} (x) + \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.5

Factor using the perfect square rule.

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Step 7.5.1

Rearrange terms.

$\frac{\sec^{2} (x) + 2 \csc (x) \sec (x) + \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \csc (x) \sec (x) = 2 \cdot \sec (x) \cdot \csc (x)$

Step 7.5.3

Rewrite the polynomial.

$\frac{\sec^{2} (x) + 2 \cdot \sec (x) \cdot \csc (x) + \csc^{2} (x)}{{(1 + \cot (x))}^{2}}$

Step 7.5.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = \sec (x)$ and $b = \csc (x)$ .

$\frac{{(\sec (x) + \csc (x))}^{2}}{{(1 + \cot (x))}^{2}}$