## Yahoo! 3 years ago Prove that cos2 2x – cos2 6x = sin 4x sin 8x

1. waterineyes

Let us start from the start..

2. Yahoo!

it is cos^2...ok

3. Mimi_x3

$cos^{2}2x -cos^{2}6x = sin4xsin8x$?

4. Yahoo!

yes

5. Mimi_x3

have you tried cos^{2}2x = (1+cos4x)/2?

6. Yahoo!

no

7. apoorvk

$\cos^2 A - \cos^2 B = \sin(A+B)\sin(B-A)$ This is a standard formula. So, we need a proof right?

8. Mitul

(1+cos4x)/2-(1+cos12x)/2 = (1+cos 4x+cos12x+cos4xcos12x)/2 and then defactorize the last term...and factorize the second and third term

9. Yahoo!

@apoorvk yup proof needed..

10. Mitul

@apoorvk it is cosA-cosBand not their squares for standard formula

11. waterineyes

$\cos^2(2x) - \cos^2(6x) \implies (\cos(2x) - \cos(6x))(\cos(2x) + \cos(6x))$ So: Using: $\cos(C) + \cos(D) = 2\cos(\frac{C+ D}{2})\cos(\frac{C-D)}{2})$ $\cos(C) - \cos(D) = 2\sin(\frac{C+ D}{2})\sin(\frac{D-C}{2})$ $(2\sin(4x) \sin(2x))(2\cos(4x)\cos(2x)) \implies (2\sin(2x)\cos(2x)) \cdot (2\sin(4x) \cos(4x))$ $\implies \sin(4x) \cdot \sin(8x)$

12. waterineyes

Using: $2\sin(x)\cos(x) \implies \sin(2x)$

13. waterineyes

I think above you by mistake written (B-A) @apoorvk

14. waterineyes

You are doing wrong I guess..

15. waterineyes

Rewrite for cos(A) - cos(B) = ??

16. Yahoo!

@apoorvk u told me it is sin(a+b).sin(b-a) but u have written differ

17. apoorvk

yeah I have memory problems -.- Amnesia *sigh* *sniff*

18. waterineyes

$\cos(A) - \cos(B) = 2\sin(\frac{A+B}{2}) \cdot \sin(\frac{B-A}{2})$

19. waterineyes

And not sin(a-B) in the last..

20. Yahoo!

@apoorvk if i use ur method we wont get that lol

21. apoorvk

NOOOOOOOO!!!!!!!!!!!!!!!!!! What have i been doing? /_\

22. apoorvk

Okay please tweak it and use it as needed, we know the procedure atleast. Or should I repost?

23. waterineyes

See: $\cos^2(2x) - \cos^2(6x) = \sin(A+B)\sin(A-B) \implies \sin(2x - 6x)\sin(2x + 6x)$ Don't you think we get negative ??? @apoorvk

24. apoorvk

yeah i got it ate already - unfortunately i can't view my raw LateX codes (some stupid bug :\), so I'll have to edit it manually. NOOOOOO!!!!!!!!!

25. apoorvk

Sir what you think is correct, if you want a confirmation. -.-

26. waterineyes

There must be condition for doing this formula : I am discussing with you @apoorvk

27. waterineyes

The above said formula will work if B > A..

28. apoorvk

$\cos^2A−\cos^2B$ $=(\cos A+\cos B)(\cos A−\cos B)$ $=[2\cos(\frac{A+B}2)\cos(\frac{A−B}2)][2\sin(\frac{A+B}2)\sin(\frac{B-A}2)]$ $=[2\sin(\frac{A+B}2)\cos(\frac{A+B}2)][2\sin(\frac{B-A}2\cos(\frac{B-A}2)]$ $=\Large\boxed{\sin(A+B)\sin(B-A)}$ Happy nitpickers? --____--

29. waterineyes

Do you think It will work ??

30. apoorvk

Water in eyes, Fish in Water $$\large \rightarrow$$ so Fish in eyes. -_-

31. waterineyes

32. waterineyes

If fish is eyes then why are looking upwards?? Is fish in your eyes??

33. apoorvk

I paarsonaaally bilib it's Phaathar bill wa-ark! -,- (dish phormula)

34. waterineyes

What is this?? @apoorvk

35. apoorvk

U know aandarishtand? oowhaat u know no inglis?? BAH HUMBUG!

36. apoorvk

Inglish pipul go, liebe children behind -_-

37. apoorvk

'Addition', man, addition ----> " sin (A '+' B) - Y U LOVE SUBTRACTION??

38. waterineyes

Okay my mistake..

39. apoorvk

U inglish phail, I inglish oowhale. -____-

40. waterineyes

You failed in english? ok I got this.. Better luck next time.. @apoorvk

41. apoorvk

Hu shaid I phail in inglish?? U no si mai bandarfool inglish haan??? o.O I gets the 105 out of 100 -__- u get how many?