How do you simplify $sin ({cos}^{- 1} (\frac{2}{3}) - {sin}^{- 1} (\frac{2}{3}))$ $sin(cos^-1(2/3)-sin^-1(2/3))$ ?

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Answer 1 · 2016-07-29T17:37:48

$\frac{1}{9}$ $1/9$

Use ${sin}^{- 1} x + {cos}^{- 1} x = \frac{π}{2}$ $sin^(-1)x+cos^(-1)x=pi/2$ . Here, $x = \frac{2}{3}$ $x = 2/3$ .

Let ${cos}^{- 1} (\frac{2}{3}) = a$ $cos^(-1)(2/3)=a$ . Then $cos a = \frac{2}{3}$ $cos a = 2/3$ .

Now, the given expression is

$sin (2 a - \frac{π}{2})$ $sin(2a-pi/2)$

$= - cos 2 a$ $=-cos 2a$

$= 1 - 2 {cos}^{2} a$ $=1-2cos^2a$

$= (1 - \frac{8}{9})$ $=(1-8/9)$

$= \frac{1}{9}$ $=1/9$

How do you simplify sin(cos^-1(2/3)-sin^-1(2/3))sin(cos−1(23)−sin−1(23))sin(cos^-1(2/3)-sin^-1(2/3))?