`"sin"^(-1)(8)/(17)+"sin"^(-1)(3)/(5)` is equal to

Options

A. `cos^(-1)((36)/(85))`
B. `cos^(-1)((17)/(38))`
C. `cos^(-1)((14)/(47))`
D. `cos^(-1)((42)/(61))`

Answer

A

Explanation

Let `sin^(-1)((8)/(17))=xandsin^(-1)((3)/(5))=y`
Then , sin x `=(8)/(17) and sin y=(3)/(5)`
Now , cos (x+y) =cos x cosy -sin xsiny
`=sqrt(1-sin^(2)x)sqrt(1-sin^(2)y)-sinxsiny`
`=sqrt(1-((8)/(17))^(2))sqrt(1-((3)/(5))^(2))-(8)/(17)xx(3)/(5)`
`=sqrt(1-(64)/(289))sqrt(1-(9)/(25))-(24)/(85)=sqrt((22)/(289))sqrt((16)/(25))-(24)/(85)=(15)/(17)xx(4)/(5)-(24)/(85)`
`rArrx+y=cos^(-1)((60)/(85)-(24)/(85))`
`rArrsin^(-1)((8)/(17))+sin^(-1)((3)/(5))=cos^(-1)((36)/(85))`