Write As An Algebraic Expression?

Question

Write each trig expression below as an algebraic expression containing x and y.

(1) sec(tan^-1 (x) + cos^-1 (y))

(2) sin(sin^-1 (x) - cos^-1 (y))

Answer 1

sec(tan^-1 (x) + cos^-1 (y))=?

Let p=tan^-1 (x) then x=tan p

Let q= cos^-1 (y) then y=cosq

sec(p+q)=1/cos(p+q)

=1/[ cosp * cosq-sinp * sinq]

=1/[1/sqrt(x^2+1) * y -x/sqrt(x^2+1) * sqrt(1-y^2)]

do the same for the other question