Sin(a+b)sin(a-b) = cos^2b - cos^2a

[zeion]

Homework Statement

Prove that: sin(a+b)sin(a-b) = cos^2b - cos^2a

Homework Equations

The Attempt at a Solution

I'm stuck.
I've tried expanding the left side, making the right side to a difference of squares, subbing the right side with sin's, dividing, factoring and all that stuff nothing gets me anywhere..

Any hints?

 

[eumyang]

After using the sum and difference identities, expand the left side. You should get a difference of two squares. One of the terms will contain a sin² b. Replace this with an expression using a well-known trig identity. Eventually, you will get the right side. If you're stuck, show us what you got.

 

[Mark44]

After you expand the left side you should have
sin²(a)cos²(b) - cos²(a)sin²(b)

Replace both sine expressions with their cosine counterparts.

 

[symbolipoint]

After trying this myself, the proof seems uncomplicated.

The lefthand side immediately becomes (sin(a)cos(b)+cos(a)sin(b))(sin(a)cos(b)-cos(a)sin(b)),
= (sin(a)cos(b))²-(cos(a)sin(b))²
=sin²a cos²b - cos²a sin²b

Next we use our most basic circular identity and set up and substitute for sin²a=1-cos²a, and similar for sin²b=1-cos²b

Giving lefthandside = (1-cos²a)cos²b-cos²a(1-cos²b)

...from which you can continue the algebra and find the righthand side expression which you expect.

 

[zeion]

After trying this myself, the proof seems uncomplicated.

For you, maybe. ~_~

Thanks guys.

 

[Mentallic]

For you, maybe. ~_~

Thanks guys.

Don't listen to him, he's being all in your face hardcore with his maths guns

 

[symbolipoint]

About #5 and #6:
At least I showed how to start and how to progress. I did not finish the algebra steps on the forum; I left those for zeion.

 

[claireashley3]

could someone show how to do the algebra? I've gotten to

(1-cos²(a))(cos²(b)) - (cos²(a))(1-cos²(b)) = (1-cos²(a)) - (1-cos²(b))

but for some reason the algebra just isn't making sense to me?

 

[eumyang]

could someone show how to do the algebra? I've gotten to

(1-cos²(a))(cos²(b)) - (cos²(a))(1-cos²(b)) = (1-cos²(a)) - (1-cos²(b))

but for some reason the algebra just isn't making sense to me?

At first glance, I'm not sure how you got the right hand side. Just distribute the left hand side.
$$(1 - \cos^2 a)(\cos^2 b) - (\cos^2 a)(1 - \cos^2 b)$$
$$= \cos^2 b - \cos^2 a \cos^2 b - \cos^2 a + \cos^2 a \cos^2 b$$
Do you see it now?

 

[SammyS]

(1-cos²(a))-(1-cos²(b))

= 1-cos²(a)-1+cos²(b)

Right?

 

[claireashley3]

ah yeah thanks sorry
looking at it now it makes sense, i messed up the right side
thank you!