Find the Exact Value cot(arcsin(u))

$\cot (\arcsin (u))$

Step 1

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

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

Step 2

Simplify the numerator.

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

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

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

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

Find the Exact Value cot(arcsin(u))

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