what is the exact value of cos(arcsin1/3-arctan1/2)?

Answer 1

a = arcsin 1/3
sin a = 1/3
cos a = sqrt(8)/3

b = arctan 1/2
tan b = 1/2
sin b = 1/sqrt(5)
cos b = 2/sqrt(5)

cos (arcsin 1/3 - arctan 1/2)
= cos (a-b)
= cos a cos b + sin a sin b
= (sqrt(8)/3)(2/sqrt(5)) + (1/3)(1/sqrt(5))
= (4sqrt(2) + 1) / 3sqrt(5)

Answer 2

previous person has used wrong formula for cos(a-b)

it is cos a cos b + sin a sin b and not cos a cos b - sin a sin b

here we go

let a= arcsin 1/3
sin a = 1/3 so cos a = sqrt(8)/3 = 2 sqrt(2)/3

b = arctan 1/2 tan b = 1/2
sin b = 1/sqrt(5) , cos b = 2/sqrt(5)

cos (arcsin 1/3 - arctan 1/2)
= cos (a-b)
= cos a cos b + sin a sin b
= 4 sqrt(2)/3sqrt(5) + 1/3sqrt(5)
= (4sqrt(2) + 1) / 3sqrt(5)