Question

Can someone show me why cos^2(x) is equal to (1/2)cos(2x) + 1/2 ?

Answer 1

no one?

Answer 2

Workin' it.

Answer 3

Gimme a mo

Answer 4

*sigh* Ignore RodneyF.

Answer 5

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

Answer 6

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?

Answer 7

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

Answer 8

There is no algebraic way? hmm..

Answer 9

There is. Its just quite fiddly

Answer 10

There is, using the sum of angles formula.

Answer 11

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)

Answer 12

Got it thanks.

Answer 13

Yep.

Answer 14

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

Answer 15

Was just about finished with mine.

Answer 16

Or doesnt that matter?

Answer 17

It assumes you know the addition of angles identity yes.

Answer 18

lol

Answer 19

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

Answer 20

oh well..

Answer 21

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.

Answer 22

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

Answer 23

im always late

Answer 24

\[cos(\theta + \phi) = cos(\theta)cos(\phi) - sin(\theta)sin(\phi)\]

So if \(\theta = \phi\) you start getting things simplified very quickly.