Find the exact value of the expression cos tan1 512 cos1 45

Find the exact value of the expression: cos [tan^-1 5/12 - cos^-1 4/5] 52/65 7/13 63/65

cos ( tan^-1(5/12) - cos^-1 (4/5) )

let x = tan^-1 (5/12)

tan x = 5/12

cos x = 12/13 , sin x = 5/13

y = cos^-1 (4/5)

cos y = 4/5 , sin y = 3/5

applying cos (x+y) formula

cos (x-y) = cos x cos y + sin x sin y

plugging the values we get

cos ( tan^-1(5/12) - cos^-1 (4/5) ) = (12/13)*(4/5) + (3/5)(5/13) = 48/65 + 15/65 = 63/65

cos ( tan^-1(5/12) - cos^-1 (4/5) ) = 63/65 (option c)