Find the Exact Value tan(arcsin(1)-arccos(1/2))

$\tan (\arcsin (1) - \arccos (\frac{1}{2}))$

Simplify each term.

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$\tan (\frac{π}{2} - \arccos (\frac{1}{2}))$

The exact value of $\arccos (\frac{1}{2})$ is $\frac{π}{3}$ .

$\tan (\frac{π}{2} - \frac{π}{3})$

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\tan (\frac{π}{2} \cdot \frac{3}{3} - \frac{π}{3})$

To write $- \frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\tan (\frac{π}{2} \cdot \frac{3}{3} - \frac{π}{3} \cdot \frac{2}{2})$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

$\tan (\frac{π \cdot 3}{2 \cdot 3} - \frac{π}{3} \cdot \frac{2}{2})$

Multiply $2$ by $3$ .

$\tan (\frac{π \cdot 3}{6} - \frac{π}{3} \cdot \frac{2}{2})$

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$\tan (\frac{π \cdot 3}{6} - \frac{π \cdot 2}{3 \cdot 2})$

Multiply $3$ by $2$ .

$\tan (\frac{π \cdot 3}{6} - \frac{π \cdot 2}{6})$

Combine the numerators over the common denominator.

$\tan (\frac{π \cdot 3 - π \cdot 2}{6})$

Simplify the numerator.

Move $3$ to the left of $π$ .

$\tan (\frac{3 \cdot π - π \cdot 2}{6})$

Multiply $2$ by $- 1$ .

$\tan (\frac{3 π - 2 π}{6})$

Subtract $2 π$ from $3 π$ .

$\tan (\frac{π}{6})$

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{\sqrt{3}}{3}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{3}$

Decimal Form:

$0.57735026 \dots$