Question

Simplify: (sin Θ − cos Θ)^2 − (sin Θ + cos Θ)^2

Answer 1

I'm going to let

x = sin Θ
y = cos Θ

to save typing and to simplify things

Answer 2

so

(sin Θ − cos Θ)^2 − (sin Θ + cos Θ)^2

turns into

(x-y)^2 - (x+y)^2

Answer 3

We have a difference of squares here. So factor to get

(x-y)^2 - (x+y)^2

((x-y)-(x+y))((x-y)+(x+y))

(x-y-x-y)(x-y+x+y)

(0x-2y)(2x+0y)

(-2y)(2x)

(-2*2)*(y*x)

-4*(x*y)

-4*sin(Θ)*cos(Θ)

Answer 4

So that means (sin Θ − cos Θ)^2 − (sin Θ + cos Θ)^2 simplifies to -4*sin(Θ)*cos(Θ)

Answer 5

that's not one of the answer choices.

Answer 6

hmm ok let me check my steps

can you post the choices?

Answer 7

a 0

b 2

c sin2 Θ

d cos2 Θ

Answer 8

Ok, so you would then simplify the sin(Θ)*cos(Θ) portion using the identity below

2*sin(Θ)*cos(Θ) = sin(2Θ)

sin(Θ)*cos(Θ) = (1/2)*sin(2Θ)

-------------------------------------------

So,

-4*sin(Θ)*cos(Θ)

turns into

-4*[(1/2)*sin(2Θ)]

(-4*/2)*sin(2Θ)

-2sin(2Θ)

Answer 9

That's as simple as I can get it, but it looks like that isn't listed either. There might be a typo? Not sure.

Answer 10

@taylor.farnsworth95

-2 Sin(2Θ)