angemnn227
Jan 7, 2009, 08:27 PM
How would I prove an Identity like the cos(2x)=1-2(sinx)^2 or something like sin(2x)=2sinx*cosx. I don't really know how to get started
galactus
Jan 8, 2009, 01:31 PM
To prove sin(2x)=2sin(x)cos(x) use the addition formula for sine.
sin(u+v)=sin(u)cos(v)+cos(u)sin(v)
Then, we can write sin(2x)=sin(x+x)
and get:
sin(x)cos(x)+cos(x)sin(x)=2sin(x)cos(x)
See there?
To prove cos(2x)=1-2sin^{2}(x)
cos(2x)=cos(x+x)=\underbrace{cos(x)cos(x)-sin(x)sin(x)}_{\text{addition formula \\for cosine}}
cos^{2}(x)-sin^{2}(x)
\underbrace{(1-sin^{2}(x))}_{\text{cos^2(x)}}-sin^{2}(x)=1-2sin^{2}(x)
Easier than you thought, huh?