## anonymous 5 years ago Can someone show me why cos^2(x) is equal to (1/2)cos(2x) + 1/2 ?

1. anonymous

no one?

Workin' it.

3. anonymous

Gimme a mo

*sigh* Ignore RodneyF.

5. anonymous

wolfram says it's true... don't know how in the world that is true.

6. anonymous

There's a few reasons you can show depending on your understanding. There's the $$e^{i\theta}$$ argument, or there's the angle addition arguement. Which would you prefer?

7. anonymous

Its basically a derivative of the double angle formula for sin and cos. I can have a go at writing the proof out, but you may be better to google it

8. anonymous

There is no algebraic way? hmm..

9. anonymous

There is. Its just quite fiddly

10. anonymous

There is, using the sum of angles formula.

11. anonymous

cos (2x) = cos^2(x) - sin^2(x) cos(2x) + sin^2(x) = cos^2(x) cos (2x) + 1 - cos^2(x) = cos^(x) cos(2x) +1 = 2cos^2(x) cos(2x)/2 + 1/2 = cos^2(x)

12. anonymous

Got it thanks.

13. myininaya

14. anonymous

Yep.

15. anonymous

But doesn't that assume the prior knowledge of that trig identity?

16. anonymous

Was just about finished with mine.

17. anonymous

Or doesnt that matter?

18. anonymous

It assumes you know the addition of angles identity yes.

19. myininaya

lol

20. anonymous

I'm in calc.. should have known that identity.. Always forget important stuff...

21. anonymous

oh well..

22. anonymous

sin 2a= sin ( a+a ) = sin a cos a + cos a sin a = 2 sin a cos a. cos 2 a= cos ( a+a ) = cos a cos a − sin a sin a cos 2a = cos² a− sin²a. . . . . . . (1) This is the first of the three versions of cos 2a. To derive the second version, in line (1) use this Pythagorean identity: sin²a = 1 − cos²a. Line (1) then becomes cos 2a = cos²a − (1 − cos²a) = cos²a − 1 + cos²a. cos 2a = 2 cos²a − 1. . . . . . . . . . (2) To derive the third version, in line (1) use this Pythagorean identity: cos² a= 1 − sin²a. We have cos 2 a= 1 − sin² a− sin²a;. cos 2a = 1 − 2 sin²a. . . . . . . . . . (3) These are the three forms of cos 2a.

23. myininaya

cos(x+x)=cos(x)cos(x)-sin(x)sin(x)

24. myininaya

im always late

25. anonymous

$cos(\theta + \phi) = cos(\theta)cos(\phi) - sin(\theta)sin(\phi)$ So if $$\theta = \phi$$ you start getting things simplified very quickly.