Another elementary proof of this is in Yaglom and Yaglom's delightful "Challenging Mathematical Problems With Elementary Solutions (Volume 2)", which is available as an inexpensive Dover edition [1].
It's problem #147. It asks you to deduce Wallis' formula from the identities that problem #143 asks you to prove. Those are:
There is also a "Challenging Mathematical Problems With Elementary Solutions (Volume 1)", also available as an inexpensive Dover edition and, unlike Volume 2, available as en ebook [2].
Volume I's problems are divided into the following categories: introductory problems; the representation of integers as sums and products; combinatorial problems on the chessboard; geometric problems on combinatorial analysis; problems on the binomial coefficients; problems on computing probabilities; experiments with infinitely many possible outcomes; and experiments with a continuum of possible outcomes.
Volume II's problems are divided into the following categories: points and lines; lattices of points in the plane; topology; a property of the reciprocals of integers; convex polygons; some properties of sequences of integers; distributions of objects; nondecimal counting; polygons with minimal deviation from zero (Tchebychev polynomials); four formulas for pi; the calculation of areas of regions bounded by curves; some remarkable limits; and the theory of primes.
These have some fun and surprising problems. For instance, problem 119 of Volume 2: Let a=1/n be the reciprocal of a positive integer. Let A and B be two points of the plane such that the segment AB has length 1. Prove that every continuous curve joining A to B has a chord parallel to AB and of length a. Show that if a is not the reciprocal of an integer, then there is a continuous curve joining A to B which has no such chord of length a.
After the problems the books provide hints, and after the hints full solutions are provided.
It's problem #147. It asks you to deduce Wallis' formula from the identities that problem #143 asks you to prove. Those are:
sin(pi/2m) sin(2pi/2m) sin(3pi/2m)...sin((m-1)pi/2m) = sqrt(m)/2^(m-1)
sin(pi/4m) sin(3pi/4m) sin(5pi/4m)...sin((2m-1)pi/4m) = sqrt(2)/2^m
There is also a "Challenging Mathematical Problems With Elementary Solutions (Volume 1)", also available as an inexpensive Dover edition and, unlike Volume 2, available as en ebook [2].
Volume I's problems are divided into the following categories: introductory problems; the representation of integers as sums and products; combinatorial problems on the chessboard; geometric problems on combinatorial analysis; problems on the binomial coefficients; problems on computing probabilities; experiments with infinitely many possible outcomes; and experiments with a continuum of possible outcomes.
Volume II's problems are divided into the following categories: points and lines; lattices of points in the plane; topology; a property of the reciprocals of integers; convex polygons; some properties of sequences of integers; distributions of objects; nondecimal counting; polygons with minimal deviation from zero (Tchebychev polynomials); four formulas for pi; the calculation of areas of regions bounded by curves; some remarkable limits; and the theory of primes.
These have some fun and surprising problems. For instance, problem 119 of Volume 2: Let a=1/n be the reciprocal of a positive integer. Let A and B be two points of the plane such that the segment AB has length 1. Prove that every continuous curve joining A to B has a chord parallel to AB and of length a. Show that if a is not the reciprocal of an integer, then there is a continuous curve joining A to B which has no such chord of length a.
After the problems the books provide hints, and after the hints full solutions are provided.
[1] https://www.amazon.com/Challenging-Mathematical-Problems-Ele...
[2] https://www.amazon.com/Challenging-Mathematical-Problems-Ele...