Question 1197232
sec(x) = 3/2 becomes:
1/cos(x) = 3/2 which becomes:
cos(x) = 2/3.
when cos(x) = 2/3, solve for sin(x) to get sin(x) = sqrt(5)/3
confirm by cos^2(x) + sin^2(x) = 1 which becomes:
(2/3)^2 + (sqrt(5)/3)^2 = 1


you have:
cos(x) = 2/3
sin(x) = sqrt(5)/3


cosec(y) = 3 becomes:
1/sin(y) = 3 which becomes:
sin(y) = 1/3
when sin(y) = 1/3, solve for cos(y) to get cos(y) = sqrt(8)/3
confirm by cos^2(y) + sin^2(y) = 1 which becomes:
(sqrt(8)/3)^2 + (1/3)^2 = 1


you have:
cos(y) = sqrt(8)/3
sin(y) = 1/3


put theme together, you have:
cos(x) = 2/3
cos(y) = sqrt(8)/3
sin(x) = sqrt(5)/3
sin(y) = 1/3


cos(x+y) = cos(x)cos(y)-sin(x)sin(y) which becomes
cos(x+y) = 2/3*sqrt(8)/3 - sqrt(5)/3*1/3 which is equal to .3800873636.


i also used my calculator to confirm this is true.
sec(x) = 3/2 becomes 1/cos(x) = 3/2 which becomes cos(x) = 2/3
cosec(y) = 3 becomes 1/sin(y) = 3 which becomes sin(y) = 1/3
arccos(2/3) = 48.1896851 degrees.
arcsin(1/3) = 19.47122063 degrees.
48.1896851 + 19.47122063 = 67.66090574 degrees.
cos(67.66090574) = .3800873636.