How to prove the trigonometric identity?

Question

#(asinalphasinbeta)^2-(acosalphacosbeta)^2+(acosalpha)^2=a^2sin^2beta#

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Answer 1 · 2018-07-14T11:09:48

Please see below.

Let #alpha=x and beta=y# then

#LHS=(asinalphasinbeta)^2-(acosalphacosbeta)^2+(acosalpha)^2#

#=(asinxsiny)^2-(acosxcosy)^2+(acosx)^2#

#=a^2sin^2xsin^2y-a^2cos^2xcos^2y+a^2cos^2x#

#=a^2sin^2xsin^2y+a^2cos^2x(1-cos^2y)#

#=a^2sin^2xsin^2y+a^2cos^2xsin^2y#

#=a^2sin^2y(sin^2x+cos^2x)=a^2sin^2y*1=a^2sin^2beta=RHS#