What is the value of tan(arcsin(2/3)-arccos(1/3)) ?

The value of tan(arcsin(2/3)-arccos(1/3)) is -(2(sqrt(5)-sqrt(2)))/3

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tan(arcsin(2/3)-arccos(1/3))

tan (arcsin (\frac{2}{3}) - arccos (\frac{1}{3}))

- \frac{2 ( 5 - 2 )}{3}

Solution steps

tan (arcsin (\frac{2}{3}) - arccos (\frac{1}{3}))

Rewrite using trig identities:

\frac{tan ( arcsin ( \frac{2}{3} ) ) - tan ( arccos ( \frac{1}{3} ) )}{1 + tan ( arcsin ( \frac{2}{3} ) ) tan ( arccos ( \frac{1}{3} ) )}

= \frac{tan ( arcsin ( \frac{2}{3} ) ) - tan ( arccos ( \frac{1}{3} ) )}{1 + tan ( arcsin ( \frac{2}{3} ) ) tan ( arccos ( \frac{1}{3} ) )}

Rewrite using trig identities:

tan (arcsin (\frac{2}{3})) = \frac{2 5}{5}

Rewrite using trig identities:

tan (arccos (\frac{1}{3})) = 22

= \frac{\frac{2 5}{5} - 2 2}{1 + \frac{2 5}{5} \cdot 2 2}

Simplify

\frac{\frac{2 5}{5} - 2 2}{1 + \frac{2 5}{5} \cdot 2 2} : - \frac{2 ( 5 - 2 )}{3}

= - \frac{2 ( 5 - 2 )}{3}

cos (30 0^{\circ}) cos (1 5^{\circ}) - sin (30 0^{\circ}) sin (1 5^{\circ})

2 cos (2 π)

\frac{0.5}{sin ( 0.5 )}

18 sin (7 3^{\circ})

(- 3 + i) (\frac{1}{8} (cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3})))

What is the value of tan(arcsin(2/3)-arccos(1/3)) ?
The value of tan(arcsin(2/3)-arccos(1/3)) is -(2(sqrt(5)-sqrt(2)))/3