Simplify:
1. sin(α+β)+sin(α-β)
2. (-8cosxsinx+4cos2x)^2 + (4cos^2x+8sinxcosx)^2
3. sqrt(sin^4xcosx) * sqrt(cos^3x)
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
√(sin^4xcosx) * √(cos^3x)
= √(sin^4x cos^4x)
= sin^2x cos^2x
= 1/4 sin2x
To simplify the given expressions, we will start by using trigonometric identities and properties of square roots. Let's solve each problem step by step:
1. Simplifying sin(α+β) + sin(α-β):
Using the sum and difference formula for sine, we have:
sin(α+β) + sin(α-β) = sinαcosβ + cosαsinβ + sinαcos(-β) - cosαsin(-β)
Since the cosine function is an even function (cos(-x) = cos(x)), and the sine function is an odd function (sin(-x) = -sin(x)), we can simplify further:
= sinαcosβ + cosαsinβ + sinαcosβ - cosαsinβ
= 2sinαcosβ, where α and β are angles.
2. Simplifying (-8cosxsinx + 4cos2x)^2 + (4cos^2x + 8sinxcosx)^2:
Expanding and simplifying each piece:
(-8cosxsinx + 4cos2x)^2 = (-8cosxsinx)^2 + 2(-8cosxsinx)(4cos2x) + (4cos2x)^2
= 64cos^2xsin^2x - 64cosxsinxcos2x + 16cos^22x
(4cos^2x + 8sinxcosx)^2 = (4cos^2x)^2 + 2(4cos^2x)(8sinxcosx) + (8sinxcosx)^2
= 16cos^4x + 64sinxcos^3x + 64sin^2x cos^2x
Adding both results together, we get:
64cos^2xsin^2x - 64cosxsinxcos2x + 16cos^22x + 16cos^4x + 64sinxcos^3x + 64sin^2x cos^2x
= 16cos^4x + 64cos^2xsin^2x + 64sin^2x cos^2x + 64sinxcos^3x - 64cosxsinxcos2x + 16cos^22x
= 16cos^2x(cos^2x + sin^2x) + 64sin^2x(cos^2x + sin^2x) + 64sinxcos^3x - 64cosxsinxcos2x + 16cos^22x
= 16cos^2x + 64sin^2x + 64sinxcos^3x - 64cosxsinxcos2x + 16cos^22x
= 16cos^2x + 64sin^2x + 64sinxcos^3x - 64cosxsinx(2cos^2x - 1) + 16cos^22x
= 16cos^2x + 64sin^2x + 64sinxcos^3x - 128cosxsin^2x + 64cos^3x + 16cos^22x
= 80cos^2x + 64sin^2x + 64sinxcos^3x - 128cosxsin^2x + 64cos^3x + 16cos^22x
Thus, (-8cosxsinx + 4cos2x)^2 + (4cos^2x + 8sinxcosx)^2 simplifies to 80cos^2x + 64sin^2x + 64sinxcos^3x - 128cosxsin^2x + 64cos^3x + 16cos^22x.
3. Simplifying sqrt(sin^4x cosx) * sqrt(cos^3x):
Applying the rules of exponents, we combine the square roots:
sqrt(sin^4x cosx) * sqrt(cos^3x) = sqrt((sin^4x cosx) * (cos^3x))
= sqrt(sin^4x cos^4x)
= sin^2x cos^2x
Thus, sqrt(sin^4x cosx) * sqrt(cos^3x) simplifies to sin^2x cos^2x.