How do you simplify $cos ({sin}^{- 1} (- \frac{3}{5}) + {cos}^{- 1} (\frac{3}{5}))$ $Cos(sin^-1 (-3/5) + cos^-1 (3/5))$ ?

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Answer 1 · 2016-05-19T15:28:55

$0, \pm \frac{24}{25}$ $0, +-24/25$ .

Let $a = {sin}^{- 1} (- \frac{3}{5})$ $a = sin^(-1) (-3/5)$ .

Then, $sin a = - \frac{3}{5} < 0$ $sin a = -3/5<0$ . So, a is in the 3rd quadrant or in the 4th.

Accordingly,

cos a = (- or +)(4/5).

Let $b = {cos}^{- 1} (\frac{3}{5})$ $b = cos^(-1) (3/5)$ .

Then, $cos b = \frac{3}{5} > 0 .$ $cos b = 3/5>0.$ So, b is in the 1st quadrant or in the 4th.

Accordingly, sin b = +- 4/5#.

Now, the given expression$ = cos ( a + b ) = cos a cos b - sin a sin b#

$= (- or +) (\frac{4}{5}) (\frac{3}{5}) - (- \frac{3}{5}) (\pm \frac{4}{5})$ $= (- or +)(4/5)(3/5)-(-3/5)(+-4/5)$

$= (- or +) (\frac{12}{25}) \pm \frac{12}{25}$ $=(- or+)(12/25)+-12/25$

$= 0, \pm \frac{24}{25}$ $=0, +-24/25$

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