Prove Identity: cos^2 2x-cos^2x=sin^2x-sin^2 2x?

Question

I loved math until i reached trigonometry. I have no idea how to simplify or prove identities. Can someone please help me? Here is the problems i need to solve

Prove the Identities:
1. cos²2x-cos² x=sin² x-sin²2x
2. 2sinx cos³ x+2sin³x cos x= sin2x
Write the expression in terms of sin x and cos x only:
3. sin2x+cos3x

Can anyone solve any one of these problems?

Answer 1

Huh?? I'm still trying to remeber my home phone number!! Now all those other numbers really got me confused!!! Now I have to start all over again!!!!!!!!!!!!!!!!!!

Answer 2

In general, you should be looking to exploit the identity cos² x + sin² x = 1 and the doucle/triple or compound angle formulae. Sometimes the method for doing this is clear, sometimes it's harder to find. Don't forget simpler techniques like factorisation, either.

1. cos² 2x - cos² x = sin² x - sin² 2x
<=> cos² 2x + sin² 2x = sin² x + cos² x
<=> 1 = 1
QED.

2. 2 sin x cos³ x + 2 sin³ x cos x = sin 2x
<=> (2 sin x cos x) (cos² x + sin² x) = sin 2x
<=> (sin 2x) (1) = sin 2x
QED.

3. sin 2x + cos 3x
= 2 sin x cos x + 4 cos³ x - 3 cos x

Note, if you don't know the triple angle formula for cos 3x, you can derive it from the double angle formulae:
cos (3x) = cos (2x + x)
= cos 2x cos x - sin 2x sin x
= (2 cos² x - 1) cos x - (2 sin x cos x) sin x
= 2 cos² x - cos x - 2 cos x sin² x
= 2 cos² x - cos x - 2 cos x (1 - cos² x)
= 2 cos² x - cos x - 2 cos x + 2 cos² x
= 4 cos³ x - 3 cos x

Answer 3

Check out the identities on
http://sosmath.com/trig/Trig5/trig5/trig5.html

www.sosmath.com You'll need to use these to solve the problem.

hint for #1.
cos² x + sin² x = 1

hint for #2.
use the double angle formulas

You should do your problems yourself so I don't want to give you the answer. Ultimately you're just hurting yourself since they're going to come up over and over again.

Buckle down and learn it right. You won't regret it. Good luck.

Answer 4

3.sin2x+cos3x=sin2x +cos2(1.5x)
=sin2x+1-2sin^2(1.5x) by one of the double angle formulas

2. 2sinxcos^3x+2sin^3xcosx
=2sinxcosx(cos^2x+sin^2x)
=2sinxcosx by the pythagorean identity
=sin2x by one of the double angle formulas