1. (-8cosxsinx+4cos2x)^2 +(4cosx^2+8sinxcosx)^2
2. cos(α-β)cosβ-sin(α-β)sinβ
3. 1+(sinx-cosx)(sinx+cosx)
4. (cosβ-cotβ)(cosβ+cotβ)
I tried these but for 4 I seemed to get a reasonable answer: cos^2(β)-cot^2(β)
How does that look? Please correct me if I'm wrong - it felt too simple.
cos^2(β)-cot^2(β) can be simplified futher.
cot = 1/tan = cos/sin
To simplify these expressions, we can use various trigonometric identities. Let's go through each one step by step:
1. (-8cosxsinx+4cos2x)^2 +(4cosx^2+8sinxcosx)^2
To simplify this expression, we can expand and combine like terms:
Expanding the first term:
(-8cosxsinx + 4cos2x)^2
= (-8cosxsinx)^2 + 2(-8cosxsinx)(4cos2x) + (4cos2x)^2
= 64cos^2xsin^2x - 64cosxsinx(2cos2x) + 16cos^22x
Expanding the second term:
(4cosx^2 + 8sinxcosx)^2
= (4cosx^2)^2 + 2(4cosx^2)(8sinxcosx) + (8sinxcosx)^2
= 16cos^4x + 64cosx^3sinx + 64cos^2xsin^2x
Adding both terms:
= 64cos^2xsin^2x - 64cosxsinx(2cos2x) + 16cos^22x + 16cos^4x + 64cosx^3sinx + 64cos^2xsin^2x
= 80cos^2xsin^2x - 64cosxsinx(2cos2x) + 16cos^22x + 16cos^4x + 64cosx^3sinx
Simplifying further is not possible without specific values for x.
2. cos(α-β)cosβ-sin(α-β)sinβ
To simplify this expression, we can use the angle difference identities:
cos(α-β)cosβ - sin(α-β)sinβ
= cosαcosβ - cosβsinαsinβ - sinαsinβcosβ
No further simplification is possible without specific values for α and β.
3. 1+(sinx-cosx)(sinx+cosx)
To simplify this expression, we will expand the square to eliminate the product of binomials:
1 + (sinx - cosx)(sinx + cosx)
= 1 + sin^2x - sinxcosx + sinxcosx - cos^2x
= 1 + sin^2x - cos^2x
Using the Pythagorean identity sin^2x + cos^2x = 1, the expression simplifies to:
= 1 + 1 - cos^2x
= 2 - cos^2x
4. (cosβ - cotβ)(cosβ + cotβ)
To simplify this expression, we can use the cotangent identity cotθ = cosθ/sinθ:
= (cosβ - cosβ/sinβ)(cosβ + cosβ/sinβ)
Finding a common denominator for the fractions:
= (sinβcosβ - cosβ/sinβ)(sinβcosβ + cosβ/sinβ)
Using the difference of squares formula, we simplify further:
= sin^2βcos^2β - cos^2β + cos^2β/sin^2β
= sin^2βcos^2β + cos^2β/sin^2β
Again, no further simplification is possible without specific values for β.