2009/002) show: arcsin[(4/√41)]+arcsin[(1/√82)=pi/4?

the things become easy in case we convert arc sin to arc tan

let x = arcsin 4/√41

so sin x = 4/√41

cos x = sqrt(1- sin ^2 x) = sqrt(1-16/41) = 5/√41

so tan x = 4/5

now let sin y = 1/√82

so cos y =√(1-1/82) = 9/√(82)

so tan y = 1/9

we heed to find tan (x+y) when tan x = 4/5 and tan y = 1/9

tan (x+y) = (tan x+ tan y)/(1-tan x tan y) = (4/5+1/9)(1-4/5*1/9) = (41/45)/(41/45) = 1

so x + y = tan ^-1 1 = pi/4 ( this is so because x < pi/4 and y < pi/4 so sum <= pi/2