`sin{2(sin^(-1)(sqrt(5))/3-cos^(-1)(sqrt(5))/3)}` is equal to `(ksqrt(5))/(81)` then `k=`

Answer

8

Explanation

Let `alpha = sin^(-1).(sqrt5)/(3)`, then `cos^(-1).(sqrt5)/(3) = (pi)/(2) - alpha`
Hence the expression becomes
`sin {2(alpha -(pi)/(2) + alpha)} = sin (4 alpha - pi)`
`= -sin 4 alpha`
`= -4 sin alpha cos alpha (1-2 sin^(2) alpha)`
`=(-4) ((sqrt5)/(3)) ((20/(3)) ((-1)/(9))`
`= (sqrt5)/(81)`