Precalc question, Help!?

Question

How do you prove:

inverse sin x + inverse cos x = pie/2 ?

(inverse sinx = arcsin x) ( inverse cosx = arccosx)

Answer 1

sin(π/2 - arccos(x)) = sin(π/2)cos(arccos(x)) - cos(π/2)sin(arccos(x)) = x - 0 = x

Take arcsine of both sides:

arcsin(sin(π/2 - arccos(x))) = arcsin(x)

π/2 - arccos(x) = arcsin(x)

π/2 = arccos(x) + arcsin(x)

arcsin(sin(θ)) ≠ θ in general; arcsin(sin(θ)) = θ only if -π/2 ≤ θ ≤ π/2, by the definition of arcsine. However, π/2 - arccos(x) is always between -π/2 and π/2 inclusive (because arccos(x) is always between 0 and π inclusive), so it is a case where the arcsine "cancels" the sine.