Question

if a+b=90°,find maximum and minimum value of sin a sinb

Answer 1

sin(b) = sin(90-a) = cos(a), using a trig identity
f(a) = sin(a)*cos(a)
This function is a maximum where df/da = 0
df/da = cos^2(a) - sin^2(a) = 0 -> cos(a) = sin(a) -> tan(a) = 1
The angle whose tangent = 1 is 45 deg.
So the maximum value of sin(a)sin(b) = sin(45)sin(45) = 1/2