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Prove that sin^(- 1)(8/17)+sin^(- 1)(3/...

Prove that `sin^(- 1)(8/17)+sin^(- 1)(3/5)=sin^(- 1)(77/85)`

A

`cos^(-1)((36)/(85))`

B

`cos^(-1)((17)/(38))`

C

`cos^(-1)((14)/(47))`

D

`cos^(-1)((42)/(61))`

Text Solution

Verified by Experts

The correct Answer is:
A
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))`
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