if cos alpha + cos twoalpha is equal to P and sin alpha + sin two alphais equal to Q then eliminate Alpha

For the sake of my convenience i am taking `alpha` as xNow we have, cosx +cos2x=P sinx+sin2x=Q Squaring and adding the two equations. Then by rearranging we get {(Sinx)^2+(cosx)^2}+{(sin2x)^2+(cos2x)^2}+2(sinx*sin2x+cosx*cos2x)=(P)^2+(Q)^2Now using the following trigonometric identities (Sink)^2+(cosk)^2=1. (Identity 1) Here k is any angle. Cos(A-B)=cosA*cosB+sinA*sinB. (2nd idn) Now going back to our question we get 1+1+2cosx=(P)^2+(Q)^2 From here we get cosx= {(P)^2+(Q)^2-2}/2 Now use it in the original equation to eliminate x.

Approved