Recursive tangent demonstrated by Mathologer Burkard Polster in "Pi is IRRATIONAL: animation of a gorgeous proof" https://www.youtube.com/watch?v=Lk_QF_hcM8A

RecursiveTangent.py

# You will need matplotlib to run this script (python -m pip install matplotlib)  
#
# Recursive tangent demonstrated by Mathologer Burkard Polster in "Pi is IRRATIONAL: animation of a gorgeous proof"
# https://www.youtube.com/watch?v=Lk_QF_hcM8A 
#
# tan(x) =                   x
#          -----------------------------------------
#           1 - x^2
#              -------------------------------------
#               3 - x^2
#                  ---------------------------------
#                   5 - x^2
#                      -----------------------------
#                       7 - x^2
#                          -------------------------
#                           9 - x^2
#                              ---------------------
#                               11 - x^2
#                                   ----------------
#                                     13 - x^2
#                                         ----------
#                                           15 - x^2
#                                               ----
#                                                ...


PI = 3.14159265359


#Defines the recursive tangent function
def tan_recursion(denominator, currLevel, maxLevel, squaredRadians):
    resulting_value = denominator
    if currLevel < maxLevel:
        resulting_value -= squaredRadians / tan_recursion(denominator+2, currLevel+1, maxLevel, squaredRadians)
    return resulting_value


#Defines the tangent function and how deep the recursion might go
def tan(radians, maxLevel):
    squaredRadians = radians*radians #calculates squared radians
    if radians == 0.0:
        return 0.0
    resulting_value = radians / tan_recursion(1, 0, maxLevel, squaredRadians)
    return resulting_value

#Defines the function that converts degrees into radians 
def degree_to_rad(angle):
    return angle*PI/180.0


#Defines the main function
def main():
    #Calculates two examples of tangent values with known values
    print("The tangent of Pi/12 (15o) is %.2f" % tan(degree_to_rad(15),10))
    print("The tangent of Pi/06 (30o) is %.2f" % tan(degree_to_rad(30),10))
    print("The tangent of Pi/04 (45o) is %.2f" % tan(degree_to_rad(45),10))
    print("The tangent of Pi/03 (60o) is %.2f" % tan(degree_to_rad(60),10))
    print("The tangent of Pi/02 (90o) is %.2f" % tan(degree_to_rad(90),10))

    #Create a list of values ranging from -180 to 175 degrees, with every 5th step (-180,-175,-170,...)
    interval_to_print = list(range(-180, 180, 5))

    #Create two lists to store the calculated tangent values
    standard_library_tangent_values = []
    recursive_tangent_tangent_values_by_depth = [[],[],[],[],[]]

    #Imports the python standard library to calculate the tangent
    import math

    #For each angle in the interval_to_print list, calculate the tangent values
    for angle in interval_to_print:
        # Transform degrees into rads
        rads = degree_to_rad(angle)

        #Calculate the tangent with the standard library
        standard_library_tangent_values.append(math.tan(rads))

        #Calculate the tangent with the recursive function with
        # varying depths of recursion (1,3,5,7,9) and store on their values
        for i in range(5):
            recursive_tangent_tangent_values_by_depth[i].append(tan(rads, (i+1)*2-1))

    #Import the graph library to plot the tangent curves
    import matplotlib.pyplot as pyplot

    #Plot the tangent curve given by the standard library
    pyplot.plot(interval_to_print, standard_library_tangent_values, 'green')

    color = ["purple","blue","orange","turquoise","red"]
    #Plot the tangent curve given by the recursive tangent functions with varying depths, each of them with a different color
    for i in range(5):
        pyplot.plot(interval_to_print, recursive_tangent_tangent_values_by_depth[i], color[i])

    #Limit the plot are to [-Pi, Pi] (X-axis) and [-10, 10] (Y-axis)
    pyplot.axis([-180, 180, -10, 10])

    #Print the plotted curves (Zoom into the converging curves to spot how close they get as the depth of recursion increases)
    pyplot.show()
    return


#Calls the main function
main()