prove that Sin x sin y sin z 14 sin xyzsin yzxsinzxy sin

prove that:

Sin x . sin y . sin z= 1/4 {sin (x+y-z)+sin (y+z-x)+sin(z+x-y) - sin (x+y+z)}

Solution

Sin x . sin y . sin z= 1/4 {sin (x+y-z)+sin (y+z-x)+sin(z+x-y) - sin (x+y+z)}

RHS 1/4 {sin (x+y-z)+sin (y+z-x)+sin(z+x-y) - sin (x+y+z)}

using sin(A+B), sin(A-B) , cos(A+B) and cos(A-B) trig identities.

= 1/4{ sin(x+y)cosz -cos(x+y)sinz + sin(y+z)cosx - cos(y+z)sinx + sin( z+x)cosy -cos(z+x)siny - sin(x+y)cosz +cos(x+y)sinz }

common terms getting cancelled

=1/4{sin(y+z)cosx - cos(y+z)sinx + sin( z+x)cosy -cos(z+x)siny}

= 1/4{ cosx( sinycosz +cosysinz) - sinx( cosycosz - sinysinz) + cosy(sinzcosx +sinxcosz) - siny( coszcosx - sinxsinz) }

common terms getting cancelled

= 1/4 { cosxcosysinz +sinxsinysinz + cosysinzcosx + sinxsinysinz}

= 1/4 (2sinxsinysiz + cosx( cosysinz +cosysinz)

= sinxsinysinz

LHS