I cannot seem to solve: cos^2(a+b)-cos^2(a-b).

Expert Answers
hala718 eNotes educator| Certified Educator

`cos^2 (a+b) - cos^2 (a-b)`

`` Factor difference between squares.

`(cos(a+b)- cos(a-b))(cos(a+b)+cos(a-b))`

`` Now use trigonometric identities:

`cos(a) + cos(b)= 2cos((a+b)/2) cos((a-b)/2) `

`cos(a)- cos(b) = -2sin((a+b)/2) sin((a-b)/2) `

`==gt cos(a+b)- cos(a-b) = -2sin((a+b+a-b)/2) sin((a+b-a+b)/2) `

`= -2sin(a) sin(b) `

`==gt cos(a+b)+cos(a-b) = 2cos((a+b+a-b)/2) cos((a+b-a+b)/2)`

`= 2cos(a) cos(b) `

`==gt cos^2 (a+b)- cos^2 (a-b)= (2cos(a)cos(b)) (-2sin(a)sin(b)) `

`= (2cos(a)sin(a) )(-2cos(b)sin(b)) `

`= sin(2a)(-sin(2b) `

`= -sin(2a) sin(2b) `

`==gt cos^2 (a+b) - cos^2 (a-b) = -sin(2a)sin(2b)`


rcmath eNotes educator| Certified Educator

Her answer correlates with your first choice. A general rule is 
So -sin(2a)sin(2b)=-2sin(a)cos(a)*2sin(b)cos(b)

berean | Student

Thank you so much!

berean | Student

Thankyou for taking the time to answer!! I guess I am confused as to how your answer correlates with the choices I have:

1.-4sin a cos a sin b cos b

2.2sin^2 a sin b cos a

3.-2sin^2 a sin b cos a

4.4sin a cos a sin b cos b

5.-tan a cot b

I guess the problem can be worked more than one way to be able to fit one of the answers.