Which of the following angles is greater? `theta_1=sin^(-1)+sin^(-1)1/3ortheta_2=cos^(-1)4/5+cos^(-1)1/3`

Explanation

`theta_(1) + theta_(2) = sin^(-1).(4)/(5) + sin^(-1).(1)/(3) + cos^(-1).(4)/(5) + cos^(-1).(1)/(3)`
`= (sin^(-1). (4)/(5) + cos^(-1).(4).(5)) + (sin^(-1).(1)/(3) + cos^(-1).(1)/(3))`
`= (pi)/(2) + (pi)/(2) = pi`
`theta_(2) = cos^(-1) [((4)/(5)) ((1)/(3)) - sqrt(1 - ((4)/(5))^(2)) sqrt(1 - ((1)/(3))^(2))]`
`= cos^(-1) [(4)/(5).(1)/(3) - (3)/(5).(sqrt8)/(3)]`
Now, `(4 - 6 sqrt2)/(15) lt 0`
Therefore, `theta_(2)` is obtuse and hence `theta_(1)` is acute.
`:. theta_(1) lt theta_(2)`