If `a^(2) + b^(2) = c^(2), c != 0`, then find the non-zero solution of the equation:
`sin^(-1).(ax)/(c) + sin^(-1).(bx)/(c) = sin^(-1) x`

Answer

`x = +- 1`

Explanation

`sin^(-1).(ax)/(c) + sin^(-1).(bx)/(c) = sin^(-1) x`
`rArr sin^(-1) ((ax)/(c) sqrt(1 - (b^(2) x^(2))/(c^(2))) + (bx)/(c) sqrt(1 - (a^(2) x^(2))/(c^(2)))) = sin^(-1) x`
`rArr (ax)/(c) sqrt(1- (b^(2) x^(2))/(c^(2))) + (bx)/(c) sqrt(1 - (a^(2) x^(2))/(c^(2))) = x`
`rArr (a)/(c) sqrt(1 - (b^(2) x^(2))/(c^(2))) + (b)/(c) sqrt(1 - (a^(2) x^(2))/(c^(2))) =1`
`rArr (a^(2))/(c^(2)) (1 - (b^(2) x^(2))/(c^(2))) + (b^(2))/(c^(2)) (1- (a^(2) x^(2))/(c^(2))) + (2ab)/(c^(2)) sqrt(1 - (a^(2) x^(2))/(c^(2))) sqrt(1- (b^(2) x^(2))/(c^(2))) = 1`
`rArr (a^(2) + b^(2))/(c^(2)) - (2a^(2) b^(2) x^(2))/(c^(4)) + (2ab)/(c^(2)) sqrt(1 - (a^(2) x^(2))/(c^(2))) sqrt(1 - (b^(2) x^(2))/(c^(2))) = 1`
`rArr sqrt(1- (a^(2) x^(2))/(c^(2))) sqrt(1 - (b^(2) x^(2))/(c^(2))) = (abx^(2))/(c^(2))`
`rArr sqrt(c^(2) - a^(2) x^(2)) sqrt(c^(2) - b^(2) x^(2)) = abx^(2)`
`rArr c^(4) - c^(2) (a^(2) + b^(2)) x^(2) + a^(2) b^(2) x^(4) = a^(2) b^(2) x^(4)`
`rArr c^(4) -c^(2) (c^(2)) x^(2) = 0`
`rArr x = +- 1`