#1:  " ----------[ D e r i v e ]---------- "                User


        #2:  " ---------- Initialization ---------- "               User


        #3:  " ---------- Trigonometry ---------- "                 User


        User

        #4:  " => - [(sqrt(5) + 1) sqrt(2)]/[(sqrt(5) - 1) sqrt(sqrt(5) + 5)]

               "


        #5:  "    = - sqrt[1 + 2/sqrt(5)]"                          User


        User

        #6:  " From B. F. Caviness, Robert P. Gilbert, Wolfram Koepf, Roman

               Shtokhamer and "


        User

        #7:  " David W. Wood, _An Introduction to Applied Symbolic

               Computation using "


        User

        #8:  " MACSYMA_, University of Delaware, draft of December 14, 1993,

               section 2.3.3. "


                / 7*pi \
        #9:  TAN|------|                                            User
                \  10  /


                   / 2*SQRT(5)     \
        #10: - SQRT|----------- + 1|                            Simp(#9)
                   \     5         /


        #11: " => - cos 3 "                                         User


                 / 1 + COS(6) \
        #12: SQRT|------------|                                     User
                 \      2     /


        #13: - COS(3)                                          Simp(#12)


        User

        #14: " cos(n pi) + sin((4 n - 1)/2 pi) => (-1)^n - 1 for integer n "


        #15: n :epsilon Integer                                     User


        #16: n                                                 Simp(#15)


                            / 4*n - 1    \
        #17: COS(n*pi) + SIN|---------*pi|                          User
                            \    2       /


                 n    
        #18: (-1)  - 1                                         Simp(#17)


        User

        #19: " cos(cos(n pi) pi) + sin(cos(n pi) pi/2) => -1 + (-1)^n for

               integer n "


                                    / COS(n*pi)*pi \
        #20: COS(COS(n*pi)*pi) + SIN|--------------|                User
                                    \       2      /


                 n    
        #21: (-1)  - 1                                         Simp(#20)


        User

        #22: " sin([n^5/5 + n^4/2 + n^3/3 - n/30] pi) => 0 for integer n "


        #23: " [Paul Zimmermann] "                                  User


                //  5      4      3        \   \
                || n      n      n       n |   |
        #24: SIN||---- + ---- + ---- - ----|*pi|                    User
                \\  5      2      3     30 /   /


                /          4       3       2      \
                | pi*n*(6*n  + 15*n  + 10*n  - 1) |
        #25: SIN|---------------------------------|            Simp(#24)
                \                30               /


        #26: n :=                                                   User


        User

        #27: " | cos x |, | sin x | => - cos x, - sin x  for  - 3 pi < x < -

               5/2 pi "


        #28: x :epsilon (- 3*pi, - 5/2*pi)                          User


        #29: x                                                 Simp(#28)

        #30: [|COS(x)|, |SIN(x)|]                                   User


        #31: [- COS(x), |SIN(x)|]                              Simp(#30)


        #32: x :=                                                   User


        User

        #33: " Trigonometric manipulations---these are typically difficult

               for students "


                   COS(3*x) 
        #34: r := ----------                                        User
                    COS(x)  


        #35: " => cos(x)^2 - 3 sin(x)^2 or similar "                User


        #36: Trigonometry := Expand                                 User


        #37: r                                                      User


                         2
        #38: 1 - 4*SIN(x)                                      Simp(#37)


        #39: " => 2 cos(2 x) - 1 "                                  User


        #40: Trigonometry := Auto                                   User


        #41: r                                                      User


        #42: 2*COS(2*x) - 1                                    Simp(#41)


        #43: " Use rewrite rules => cos(x)^2 - 3 sin(x)^2 "         User


        #44: r :=                                                   User


        #45: " Here is a tricky way of writing 0/0 "                User


                            2             2 
                      TAN(x)  + 1 - SEC(x)  
        #46: expr := -----------------------                        User
                            2         2     
                      SIN(x)  + COS(x)  - 1 


        #47: " Let's try simplifying this expression! "             User


        #48: expr                                                   User
        #49: ?                                                 Simp(#48)


        #50: " What is its limit at zero? "                         User


             lim  expr
        #51: x->0                                                   User


        #52: ?                                                 Simp(#51)


        #53: " What is the derivative? "                            User


             d      
        #54: -- expr                                                User
             dx     


        #55: ?                                                 Simp(#54)


        #56: expr :=                                                User


        #57: " ---------- Quit ---------- "                         User