Precalc and trig
posted by hayden on .
cos^2 (x+a)sin^2(a)= (cos x)(cos (2a+x))
prove using only side i worked on the left side...

I worked on both sides,
LS = (cosxcosa – sinxsina)^2 – sin^2a
= cos^2xcos^2a – 2cosxcosasinxsina + sin^2xsin^2a – sin^2a
= cos^2xcos^2a – 2cosxcosasinxsina + sin^2a(sin^2x – 1)
= cos^2xcos^2a – 2cosxcosasinxsina – sin^2acos^2x
= cos^2x(cos^2a – sin^2a) – 2cosxcosasinxsina
RS
= cosx[cos2acosx – sin2asinx]
= cosx[(cos^2a – sin^2a)cosx – 2sinacosasinx]]
= cosx[cos^2acosx – sin^2acosx – 2sinacosasinx]
= cos^2acos^2x – sin^2acos^2x  2sinacosasinx
= cos^2x(cos^2a – sin^2a)  2sinacosasinx
= LS
Q.E.D. 
thanks! i accidentally posted this twice

how does
cos^2x(cos^2a – sin^2a)  2sinacosasinx
=
cos^2x(cos^2a – sin^2a) – 2cosxcosasinxsina 
thanks for catching that typo, it is so hard to type those complicated trig expressions without making some errors.
on RS it looks like I forgot to multiply cosx by that last term
so from
cosx[cos^2acosx – sin^2acosx – 2sinacosasinx]
= cos^2acos^2x – sin^2acos^2x  2sinacosasinxcosx
= cos^2x(cos^2a – sin^2a)  2sinacosasinxcosx
= LS