sin(θ+φ+ψ)=sinθcosφcosψ+cosθsinφcosψ+cosθcosφsinψ-sinθsinφsinψ

Question

#22, page 126 in Plane and Spherical Trigonometry by Paul Rider

Answer 1

Replace Greek symbols with A, B, C.

sin(A+B+C)=sinAcos(B+C)+cosAsin(B+C)=

sinA(cosBcosC-sinBsinC)+cosA(sinBcosC+cosBsinC)=

sinAcosBcosC-sinAsinBsinC+cosAsinBcosC+cosAcosBsinC.

This is the same as sinAcosBcosC+cosAsinBcosC+cosAcosBsinC-sinAsinBsinC.

Replace A, B, C with original Greek symbols.