cos^2 (x+a)-sin^2(a)= (cos x)(cos (2a+x))

Question

prove using only side i worked on the left side...

Answer 1

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.

Answer 2

thanks! i accidentally posted this twice

Answer 3

how does

cos^2x(cos^2a – sin^2a) - 2sinacosasinx

=

cos^2x(cos^2a – sin^2a) – 2cosxcosasinxsina

Answer 4

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