# Deriving identities

• Aug 22nd 2007, 11:24 AM
ocmisssunshine
Deriving identities
Okay these are giving me such a problem to figure out. I have to use these 4 identities to prove others.
A. sin(-x) = - sinx
B. cos(-x) = -cosx
C. cos(x+y) = cosxcosy-sinxsiny
D. sin(x+y) = sinxcosy + cosxsiny

these are what i have to derive.

cos2x=1-2sin^2x

Abslout value of cos(x/2) = Squareroot of ((1+cosx)/2)

Abslout value of sin(x/2) = Squareroot of ((1-cosx)/2)

Any help is much apprecieted. Thanks alot!
• Aug 22nd 2007, 01:10 PM
CaptainBlack
Quote:

Originally Posted by ocmisssunshine
Okay these are giving me such a problem to figure out. I have to use these 4 identities to prove others.
A. sin(-x) = - sinx
B. cos(-x) = -cosx
C. cos(x+y) = cosxcosy-sinxsiny
D. sin(x+y) = sinxcosy + cosxsiny

these are what i have to derive.

cos2x=1-2sin^2x

Use C with $\displaystyle y=x$:

$\displaystyle \cos(x+y)=\cos(2x) = \cos^2(x)-\sin^2(x)$

Now use $\displaystyle \cos^2(x)=1-\sin^2(x)$ to get:

$\displaystyle \cos(2x) = \cos^2(x)-\sin^2(x)=1-2\sin^2(x)$

RonL
• Aug 22nd 2007, 01:17 PM
CaptainBlack
Quote:

Originally Posted by ocmisssunshine
Okay these are giving me such a problem to figure out. I have to use these 4 identities to prove others.
A. sin(-x) = - sinx
B. cos(-x) = -cosx
C. cos(x+y) = cosxcosy-sinxsiny
D. sin(x+y) = sinxcosy + cosxsiny

these are what i have to derive.

cos2x=1-2sin^2x

Abslout value of cos(x/2) = Squareroot of ((1+cosx)/2)

The partner of the previous identity is:

$\displaystyle \cos(2x)=1+2\cos^2(x)$

put $\displaystyle u=2x$, so:

$\displaystyle \cos(u)=1+2\cos^2(u/2)$

Now rearrange:

$\displaystyle |\cos(u/2)| = \sqrt{(1+\cos(u))/2}$

Now replace $\displaystyle u$ with $\displaystyle x$:

$\displaystyle |\cos(x/2)| = \sqrt{(1+\cos(x))/2}$

Quote:

]
Abslout value of sin(x/2) = Squareroot of ((1-cosx)/2)

Any help is much apprecieted. Thanks alot!
Do this like the one above but using the result of the first part

RonL