Question

Prove that `sin^-1  8/17 + sin^-1  3/5 = sin^-1  7/85`

Answer 1

L.H.S. `sin^-1  8/17 + sin^-1  3/5`

Using `sin^-1x +sin^-1y  sin^-1[xsqrt(1 - y^2) + ysqrt(1 - x^2)]`

`sin^-1  8/17 + sin^-1  3/5 = sin^-1[8/17* sqrt(1 - (3/5)^2) + 3/5 * sqrt(1  (8/1)^2)]`

= `sin^-1[8/17 * sqrt(1  9/25) + 3/5* sqrt(1 - 64/289)]`

= `sin^-1 [8/17 * sqrt(16/25) + 3/5* sqrt(225/289)]`

= `sin^-1 [8/17 * 4/5 +3/5 * 15/17]`

= `sin-1 [32/85 + 45/85]`

=`sin^-1  77/85` R.H.S.

Hence proved.