## sabika13 3 years ago how do prove this identity: sin^4x -cos^4x=1-2cos^2x

1. zordoloom

Do you still need help?

2. sabika13

RS: (sin^2x+cos^2x)-2cos^2x sin^2x-cos^2x Thats how much i can simplify right side..

3. sabika13

yah

4. zordoloom

alright.

5. zordoloom

So here is how I proved it:

6. zordoloom

So, change sinx^4 to (1-cosx^2)^2

7. zordoloom

You can expand that and you get 1-2cosx^2+cosx^4-cosx^4=1-2cosx^2

8. zordoloom

Do you follow so far?

9. zordoloom

@sabika13 Should I continue?

10. sabika13

yes sorry

11. zordoloom

ok

12. zordoloom

When you simplify 1-2cosx^2+cosx^4-cosx^4=1-2cosx^2, the left side becomes 1-2cosx^2=1-2cosx^2

13. zordoloom

That proves the identity.

14. zordoloom

Normally, when you prove an identity, you work on one side. In this problem I worked on the left side.

15. sabika13

wait how does (1-2cosx^2)^2=1-2cos^2+cosx^4?

16. zordoloom

To summarize it: You have sin(x)^4-cos(x)^4=1-2cos(x)^2 change sin(x)^4 --->(1-cos(x)^2)^2 You have: (1-cos(x)^2)^2-cos(x)^4=1-2cos(x)^2 Then expand (1-cos(x)^2)^2 on the left side. You get 1-2cos(x)^2+cos(x)^4-cos(x)^4=1-2cos(x)^2

17. zordoloom

Any better?

18. zordoloom

Then the: cos(x)^4-cos(x)^4 cancel out.

19. zordoloom

Hope that helped.

20. sabika13

im having problems expanding (1-cos^2x)^2 shouldnt it be: 1-cos^4x :S

21. zordoloom

So, (1-cos^2x)(1-cos^2x). Just foil out.

22. sabika13

ohhh okay, but why is it not: (1-cos^2x)(1+cos^2x)

23. zordoloom

It's squared. It's not a difference of squares.

24. sabika13

ohh i think i get it...

25. zordoloom

Any number, or equation squared is the same as it times by itself. Example (6)^2= (6)*(6) or (3x-2)^2= (3x-2)(3x-2). Do you get the point?

26. zordoloom

You don't change the signs inside the equation.

27. sabika13

yeah so whats difference of squares then (if you dont mind me asking)

28. zordoloom

example x^2-4. When you factor it, you get (x-2)(x+2). It's a^2-b^2

29. sabika13

ohhh thankkyou!!you helped me fix a very big misconception inmy head ;P

30. zordoloom

yep, no prob.