Question

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

Options include:
-sin^2 Θ
-cos^2Θ
0
2

Answer 1

So what would you get should you foil out both of those?

Answer 2

Ok I'm going to let sin Θ = x and cos Θ = y

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

(x+y)(x+y)
(x^2 +xy+y^2)

Answer 3

so (sin Θ^2 - sin Θ times cos Θ + cos Θ^2) + (sin Θ^2 + sin Θ times cos Θ + cos Θ^2)

but what is sin theta times cos theta??

Answer 4

Well, you should have a \(-2\sin(\theta)\cos(\theta)\) and a \(+2\sin(\theta)\cos(\theta)\), so those would cancel anyway, we don't need to worry about them.

Answer 5

ohh I forgot the outside values

Answer 6

But once you cancel and combine everything, you would have
\(2\sin^{2}(\theta) + 2\cos^{2}(\theta) = 2(\sin^{2}(\theta) + \cos^{2}(\theta))\)

Answer 7

Okay so it should be (sin^2 theta + cos^2 theta) + (sin^2 theta + cos^2 theta)?
And the pythagorean identity says that sin^2 theta + cos^2 theta = 1, so 1 + 1 = 2?

Answer 8

Yep :)

Answer 9

Thank you so much!

Answer 10

Yep, no problem :3

Answer 11

i have seen this question here over and over lol
did teacher run out of problems or something hehe

Answer 12

teachers*