Evaluate: `cos(sin^(-1)3/5+sin^(-1)5/(13))`

Explanation

Let `sin^(-1)=3/5=A` and `sin^(-1)5/13=B`. Then,
`A,B in [-pi/2,pi/2] rArr cosA gt 0` and `cosB gt 0`.
`therefore sinA=3/5` and `sinB=5/13`
`rArr cosA=sqrt(1-sin^(2)A)=sqrt(1-9/25)=sqrt(16/25)=4/5` and `cosB=sqrt(1-sin^(2)B) = sqrt(1-25/169)=sqrt(144/169)=12/13`.
`therefore cos(sin^(-1)3/5+sin^(-1)5/13)=cos(A+B)=cosAcosB-sinAsinB`
`=(4/5 xx 12/13) -(3/5 xx 5/13)`
`=(48/65 -15/65) = 33/65`.