Question

Derive this identity from the sum and difference formulas for cosine: sin a sin b = (1 / 2)[cos(a – b) – cos(a + b)]

Answer 1

Start with the right hand side because it is more complex

Answer 2

@tkhunny please help

Answer 3

Please write down the sum and difference formulas for cosine.

Answer 4

Cos(a + b) = cos a cos b + sin a sin b
sin (a + b) = sin a cos b + sin b cos a

Answer 5

No, that's the sum formula for sine and cosine. Write down the sum and difference formulas for cosine. We don't want any such thing for sine.

Actually, that's not the sum formula for cosine, either.

Look at the cosine formulas. Look at them very carefully. Get them right.

Answer 6

cos(a-b) = cosA cosB + sinA sinB
cos(a+b) = cosA cosB - sinA sinB ?

Answer 7

That's better. Question: What will happen if we just add these two equations?

Answer 8

You'd get 0. Don't they cancel each other out?

Answer 9

No. You're guessing. You need to start proving. If we add them, we get...

cos(a-b) + cos(a+b) = 2cos(A)cos(b)

The parts with the sine do go away.

Reassess. How close are we? Are we going the right way? If not, can we fix it?

Answer 10

Well it looks like we are getting closer. We have cos(a-b) + cos(a+b) which is similar to the cos(a-b) - cos(a+b)

Answer 11

It's a good try, but we need the sine portions, not the cosine portions. What happens if we SUBTRACT, rather than add?

Answer 12

Truthfully I'm not sure. I know the beginning will be cos(a-b) - cos (a+b) = …

Answer 13

Right, then the cosine pieces go away. What happens with the sine pieces?

Answer 14

2sin(a)sin(b)

Answer 15

or -2

Answer 16

cos(a-b) = cos(a) cos(b) + sin(a) sin(b)
cos(a+b) = cos(a) cos(b) - sin(a) sin(b)
Subtracting
cos(a-b) - cos(a+b) = 2sin(a) sin(b)

Reassess. How close are we? Are we going the right way? If not, can we fix it?

Answer 17

I believe we are because now we have the cos(a-b) - cos(a+b) which is needed

Answer 18

@niksva pls help

Answer 19

@washcaps, @tkhunny has explained this problem really well

Answer 20

but I still don't know how to fill in the table which is the question

Answer 21

@niksva

Answer 22

we are using the formulae for cos(A+B) and cos(A-B), so that can be put in the reasons group
as far as the calculation is concerned , we have just done the subtraction
i dont think we need to give the proof of cos(A+B) or other one

Answer 23

so. 1. I will show what I will subtract ---> reason use formula cos(a+b) and cos(a-b)?

Answer 24

That doesn't sound like the right reason @niksva

Answer 25

this is mathematics we use formulae
if u want to give more reasons , then give them the proof of cos(A+b) and cos(A-B)

Answer 26

This would be easier if you could lay it out all in a table. I'm getting quite confused. I would really appreciate it @niksva

Answer 27

Sorry,i could have but it is not allowed here (as per the rules of this forum)
people are here to help only
u have to do your work

Answer 28

Well could you explain a little better. I'm not asking for you to hand me the answers. I want to see how you actually got the answers. I know the first calculation is the subtracting, but I don't know the reason or what comes after @niksva

Answer 29

if i had known the better way, i would have explained it to you
@tkhunny has explained all the steps to you really well
there is nothing which I can add to his explanation

Answer 30

I only see why step that @tkhunny explained. There are supposed to be 3

Answer 31

not why. I meant one

Answer 32

i will write the steps,
cos(a-b) = cosa*cosb+sina*sinb ------(1)
cos(a+b) =cosa*cosb - sina*sinb -------(2)
(1) - (2)
cos(a-b) - cos(a+b) = 2sina*sinb
divide both sides by 2
(1/2)*{cos(a-b)-cos(a+b)} = sina*sinb

Answer 33

so 1. cos(a-b) = cosa*cosb+sina*sinb -----> reason: use cosine sums and differences
cos(a+b) =cosa*cosb - sina*sinb
2. cos(a-b) - cos(a+b) = 2sina*sinb ----> reason: subtracted difference
3. (1/2)*{cos(a-b)-cos(a+b)} = sina*sinb ----> reason: ????

@niksva

Answer 34

divide both sides by 2

Answer 35

Does this seem like a correct answer to fill in the table with?

Answer 36

yeah it is

Answer 37

Thank you! I hope it is right lol