Question

Will METAL!!

Verify the identity.

cos (x - y) - cos (x + y) = 2 sin x sin y

Answer 1

we'll be using two rules here:

cos(A+B) = cosA*cosB - sinA*sinB
cos(A-B) = cos(A)*cos(B) + sin(A)*sin(B)

Answer 2

cos(x-y) = cosx*cosy + sinx*siny
cos(x+y) = cosx*cosy - sinx*siny

therefore:

cos(x-y) - cos(x+y) = cosx*cosy + sinx*siny - (cosx*cosy - sinx*siny)

= 2sinx*siny

let me know if anything needs to be clarified

Answer 3

wait could you break that down a bit.

Answer 4

sure.

start with

cos (x - y) = cosx*cosy + sinx*siny

is that clear?

Answer 5

yes

Answer 6

cos (x + y) = cosx*cosy - sinx*siny

is that clear?

Answer 7

yes

Answer 8

good, now we just subtract

cos(x-y) - cos(x+y) = cosx*cosy + sinx*siny - (cosx*cosy - sinx*siny)

is this part clear?

Answer 9

alright what's next after that?

Answer 10

just basic algebra after that

Answer 11

notice how the cosines cancel out

Answer 12

oh ok, could we do.
Using the given zero, find all other zeros of f(x).

-2i is a zero of f(x) = x4 - 21x2 - 100

Answer 13

@freckles having a bit of a brain fart about complex zeros

Answer 14

lol ok thats fine. one min brb

Answer 15

I remember that complex zeros come in pairs, so if -2i is a zero, then +2i must also be a zero

Answer 16

so, we can write an expression using each root

x + 2i = 0
x - 2i = 0

then I'm guessing we multiply these two together to get

(x+2i)(x-2i) = 0

x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4 as our polynomial

Answer 17

then we take our original function

f(x) = x4 - 21x2 - 100

divide by the polynomial we just got

x^2 + 4

using synthetic division, I presume

Answer 18

(it's been so long since I've done this lol)