Trigonometry Examples

$\cos (arccsc (u))$

Step 1

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

$\frac{\sqrt{u^{2} - 1}}{u}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

$\frac{\sqrt{u^{2} - 1^{2}}}{u}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = u$ and $b = 1$ .

$\frac{\sqrt{(u + 1) (u - 1)}}{u}$