anonymous one year ago Verify the identity. cos(4u) = cos^2(2u) - sin^2(2u)

1. anonymous

$\cos \left( 4u\right) = \cos ^{2}\left( 2u \right) - \sin ^{2}\left( 2u \right)$

2. anonymous

@Luigi0210

3. anonymous

@Hero

4. anonymous

@arindameducationusc

5. anonymous

@kaitlyn_nicole

6. anonymous

@kaitlyn_nicole

7. anonymous

@Michele_Laino

8. arindameducationusc

Do you know Cos2theta= cos^2theta-sin^2theta?

9. anonymous

yes, double angle identity

10. Michele_Laino

hint: we can rewrite the left side as below: $\Large \cos \left( {4u} \right) = \cos \left( {2u + 2u} \right)$

11. Michele_Laino

now you can apply the formula of addition for cosine function

12. Michele_Laino

$\Large \cos \left( {x + y} \right) = \cos x\cos y - \sin x\sin y$

13. anonymous

cos ( 2u + 2u ) = cos (2u) cos (2u) - sin (2u) sin (2u) ?

14. Michele_Laino

that's right!

15. anonymous

what do i do from there?

16. Michele_Laino

it is simple, since: $\Large \cos \left( {2u} \right)\cos \left( {2u} \right) = {\left( {\cos \left( {2u} \right)} \right)^2}$

17. Michele_Laino

similarly for sin(2u)*sin(2u)

18. anonymous

sin(2u) sin(2u) = (sin(2u))^2 ?

19. Michele_Laino

yes!

20. anonymous

okay, so since im trying to verify this identity, what else do i do to get the answer to look like cos(4u) = cos(4u) ?

21. Michele_Laino

since we have proven that left side is equal to right side, then we are done here

22. anonymous

so is the final answer cos ( 2u + 2u ) = cos (2u) cos (2u) - sin (2u) sin (2u) or cos(2u) cos(2u) = (cos(2u))^2 ?

23. Michele_Laino

the answer is the entire procedure which shows how to get the right side, starting from left side

24. anonymous

ahhhhhh, i see, i see. thank you soooooo much for your help!

25. Michele_Laino

:)