## treefriendly 2 years ago Can someone help me prove sin^2(x)cos^2(x) = (1-cos4x)/theta? I'm having trouble with trigonometry and I can't really grasp the material. Please and thank you

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1. hartnn

$$\cos 2x = 2\cos^2x-1 = 1-2\sin^2x$$ so can you find cos^2x =..... ? and sin^2x =.... from this formula ?

2. treefriendly

Not really :/

3. hartnn

$$\cos 2x = 2\cos^2x-1$$ Add 1 to both sides, what u get ?

4. treefriendly

1 + cos2x = 2 cos^2 x?

5. hartnn

thats correct, now divide both sides by 2 what u get ?

6. treefriendly

(1+cos2x)/2 = cos^2 x?

7. hartnn

yes, correct. in the same way, isolate sin^2 x from $$\cos 2x = 1-2\sin^2x$$

8. treefriendly

so then cos2x-1 = -2sin^2 x?

9. hartnn

yes, now divide both sides by -2 what u get ?

10. treefriendly

(cos2x-1)/-2 = sin^2 x?

11. treefriendly

but I thought you couldn't divide by -2 because it's with the sine?

12. hartnn

yes, or you can say, (1-cos 2x)/2 = sin^2 x you can divide, there's no problem with that. now substitute both these results in left side of your equation sin^2x . cos^2x =... ?

13. treefriendly

what happened to the negative sign that was on the 2?

14. hartnn

cos2x -1 got turned into 1-cos2x because of that negative sign.

15. treefriendly

ohh okay. so ((1-cos^2 x)/2) ((1+cos2x)/2)) ?

16. hartnn

how cos^2 x ? aren't both of them cos 2x ?

17. treefriendly

Darn. Yeah, you're right. I was looking at something else when I was typing

18. hartnn

so, ((1-cos2 x)/2) ((1+cos2x)/2)) now use $$(a-b)(a+b)=a^2+b^2$$ so, (1-cos2x)(1+cos2x) =... ?

19. treefriendly

1+cos2x-cos2x-2cos^2 x

20. hartnn

check your last term. its not -2cos^2x

21. treefriendly

errr..... -cos^2 x^2? or -cos^2 2x?

22. hartnn

-cos^2 2x

23. hartnn

so, you have 1-cos^2 2x according to $$\cos 2x = 2\cos^2x-1$$ what will be cos^2 2x=... ?)

24. treefriendly

cos2x?

25. hartnn

$$\cos 2x = 2\cos^2x-1 \\cos 2(2x) = 2\cos^2(2x)-1 \\ \cos^22x =(\cos 4x+1)/2$$ got this ?

26. treefriendly

kind of yeah

27. hartnn

so, whats $$\\~ \\ 1-\cos^22x =1-(\cos 4x+1)/2=..?$$

28. treefriendly

cos 2 (2x) = cos ^2 (2x) ?

29. hartnn

ou had $$((1-\cos2 x)/2) (1+\cos2x)/2))= (1-\cos^2 2x)/4 \\ =(2-\cos 4x-1)=1-\cos 4x$$

30. AravindG

@treefriendly got it now?

31. treefriendly

Sort of, but I'm a bit lost.

32. AravindG

can you tell me where?

33. treefriendly

The last part. What happened to the 8 from the initial equation?

34. AravindG

i dont see any 8 here

35. treefriendly

Sorry, it's getting late and I've been misreading

36. AravindG

i see ..then i suggest you take rest do it afresh the next day

37. treefriendly

I meant what happened to theta that was being divided from 1-cos4x?

38. AravindG

there is no theta here

39. treefriendly

prove sin^2(x)cos^2(x) = (1-cos4x)/theta?

40. hartnn

its not theta, its 8 $$((1-\cos2 x)/2) (1+\cos2x)/2))= (1-\cos^2 2x)/4 \\\\=(1-(\cos4x+1)/2)/4\\ =(2-\cos 4x-1)/8=(1-\cos 4x)/8$$