Precalculus Examples

$\csc (\arctan (\frac{1}{x}))$

Step 1

Draw a triangle in the plane with vertices $(1, \frac{1}{x})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{1}{x})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{1}{x})$ . Therefore, $\csc (\arctan (\frac{1}{x}))$ is $\frac{\sqrt{1 + {(\frac{1}{x})}^{2}}}{\frac{1}{x}}$ .

$\frac{\sqrt{1 + {(\frac{1}{x})}^{2}}}{\frac{1}{x}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{1 + {(\frac{1}{x})}^{2}} x$

Step 3

Simplify the expression.

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Step 3.1

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

$\sqrt{1 + \frac{1^{2}}{x^{2}}} x$

Step 3.2

One to any power is one.

$\sqrt{1 + \frac{1}{x^{2}}} x$

Step 3.3

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{x^{2}}{x^{2}} + \frac{1}{x^{2}}} x$

Step 4

Combine the numerators over the common denominator.

$\sqrt{\frac{x^{2} + 1}{x^{2}}} x$

Step 5

Rewrite $\frac{x^{2} + 1}{x^{2}}$ as ${(\frac{1}{x})}^{2} (x^{2} + 1)$ .

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Step 5.1

Factor the perfect power $1^{2}$ out of $x^{2} + 1$ .

$\sqrt{\frac{1^{2} (x^{2} + 1)}{x^{2}}} x$

Step 5.2

Factor the perfect power $x^{2}$ out of $x^{2}$ .

$\sqrt{\frac{1^{2} (x^{2} + 1)}{x^{2} \cdot 1}} x$

Step 5.3

Rearrange the fraction $\frac{1^{2} (x^{2} + 1)}{x^{2} \cdot 1}$ .

$\sqrt{{(\frac{1}{x})}^{2} (x^{2} + 1)} x$

Step 6

Pull terms out from under the radical.

$\frac{1}{x} \sqrt{x^{2} + 1} x$

Step 7

Combine $\frac{1}{x}$ and $\sqrt{x^{2} + 1}$ .

$\frac{\sqrt{x^{2} + 1}}{x} x$

Step 8

Cancel the common factor of $x$ .

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Step 8.1

Cancel the common factor.

$\frac{\sqrt{x^{2} + 1}}{x} x$

Step 8.2

Rewrite the expression.

$\sqrt{x^{2} + 1}$