trigonometric-identities.egi

(coefficients (* (+ (cos α) (* i (sin α))) (+ (cos β) (* i (sin β))))
              i)
;{(+ (* (cos α) (cos β)) (* -1 (sin α) (sin β))) (+ (* (cos α) (sin β)) (* (sin α) (cos β)))}

;(cos (+ α β)) = (+ (* (cos α) (cos β)) (* -1 (sin α) (sin β)))
;(sin (+ α β)) = (+ (* (cos α) (sin β)) (* (sin α) (cos β)))


(coefficients (* (+ (cos α) (* i (sin α))) (- (cos β) (* i (sin β))))
              i)
;{(+ (* (cos α) (cos β)) (* (sin α) (sin β))) (+ (* -1 (cos α) (sin β)) (* (sin α) (cos β)))}

;(cos (- α β)) = (+ (* (cos α) (cos β)) (* (sin α) (sin β)))
;(sin (- α β)) = (+ (* -1 (cos α) (sin β)) (* (sin α) (cos β)))


(coefficients (+ (* (+ (cos α) (* i (sin α))) (+ (cos β) (* i (sin β))))
                 (* (+ (cos α) (* i (sin α))) (- (cos β) (* i (sin β)))))
              i)
;{(* 2 (cos α) (cos β)) (* 2 (sin α) (cos β))}

;(* (cos α) (cos β)) = (* (/ 1 2) (+ (cos (+ α β)) (cos (- α β))))
;(* (sin α) (cos β)) = (* (/ 1 2) (+ (sin (+ α β)) (sin (- α β))))


(coefficients (- (* (+ (cos α) (* i (sin α))) (+ (cos β) (* i (sin β))))
                 (* (+ (cos α) (* i (sin α))) (- (cos β) (* i (sin β)))))
              i)
;{(* -2 (sin α) (sin β)) (* 2 (cos α) (sin β))}

;(* (sin α) (sin β)) = (* (/ -1 2) (- (cos (+ α β)) (cos (- α β))))
;(* (cos α) (sin β)) = (* (/  1 2) (- (sin (+ α β)) (sin (- α β))))


(coefficients (** (+ (cos α) (* i (sin α))) 3)
              i)
;{(+ (cos α)^3 (* -3 (cos α) (sin α)^2)) (+ (* 3 (cos α)^2 (sin α)) (* -1 (sin α)^3))}

;(cos (* 3 α)) = (+ (cos α)^3 (* -3 (cos α) (sin α)^2))
;              = (+ (* 4 (cos α)^3) (* -3 (cos α)))
;(sin (* 3 α)) = (+ (* 3 (cos α)^2 (sin α)) (* -1 (sin α)^3))
;              = (+ (* -4 (sin α)^3 (* 3 (sin α))))