Question

Simplify:
1. sin(α+β)+sin(α-β)

2. (-8cosxsinx+4cos2x)^2 + (4cos^2x+8sinxcosx)^2

3. sqrt(sin^4xcosx) * sqrt(cos^3x)

Answer 1

sin(α+β)+sin(α-β)
= sinα cosβ + cosα sinβ - (sinα cosβ - cosα sinβ)
= 2cosα sinβ

I suspect a typo. I will use
(-8cosxsinx+4cos2x)^2 + (4cos2x+8sinxcosx)^2
= (-4sin2x+4cos2x)^2 + (4cos2x+4sin2x)
= 4^2((cos2x-sin2x)^2 + (cos2x+sin2x)^2)
assuming 2x for brevity,
= 16(cos^2-2sin*cos+sin^2)+(cos^2+2sin*cos+sin^2))
= 16(1-2sin*cos+1+2sin*cos)
= 16(2)
= 32

Answer 2

√(sin^4xcosx) * √(cos^3x)
= √(sin^4x cos^4x)
= sin^2x cos^2x
= 1/4 sin2x