Question

Show that `sin^(-1)""(3)/(5)-sin^(-1)""(8)/(17)=cos^(-1)""(84)/(85)`

Answer 1

Let `sin^(-1)3/5=x` and `sin^(-1)8/17=y`,Then,
`sinx=3/5` and `siny=8/17`.
`therefore cosx=sqrt(1-sin^(2)x)=sqrt(1-9/25)=sqrt(16/25)=4/5`
and `cosy=sqrt(1-sin^(2)y)=sqrt(1-64/25)=sqrt(225/289)=15/17`.
`therefore cos(x-y)=cosxcosy+sinxsiny`
`=(4/5 xx 15/17) +(3/5 xx 8/17) = (12/17+24/85)=84/85`
`rArr x-y=cos^(-1)(84/85) rArr sin^(-1)3/5-sin^(-1)8/17=cos^(-1)84/85`.