Find the exact value of the expression Costan1 5 cscarctan84

Find the exact value of the expression. Cos(tan^-1 5) csc[arctan(-84/13)]

Solution

cos(tan^-1(5))

Let tan^-1(5) =x

tan x=5

Here opposite=5 and adjacent=1

Therefore hypotenuse=sqrt(5^2 + 1^2)=sqrt26

cos x=adjacent/hypotenuse= 1/sqrt26=sqrt26/26

Hence,cos (tan^-1(5))= sqrt26/26

csc(tan^-1(-84/13))

Let tan^-1(-84/13)=x

tanx=-84/13

Here opposite=84 and adjacent=-13

Therefore hypotenuse=sqrt((-84)^2 + 13^2)= sqrt7225=85

And csc x= hypotenuse/opposite= 85/84

Therefore csc(tan^-1(-84/13))=85/84