User:Wonghang

It is User:wonghang.

=Pi formula proofs=

Proof of Leibniz's formula
Proof of Leibniz formula

Proof of Wallis's product
Proof of Wallis product

Proof of &zeta;(2)
Basel problem

Proof of Machin's formula
Machin's formula
 * $$\frac{\pi}{4} = 4 \tan^{-1} \frac{1}{5} - \tan^{-1} \frac{1}{239}$$

Proof
Recall the formulas:


 * $$\tan (x+y) = \frac{\tan x + \tan y}{1- \tan x \tan y}$$
 * $$\tan (x-y) = \frac{\tan x - \tan y}{1+ \tan x \tan y}$$
 * $$\tan (2x) = \frac{2 \tan x}{1 - \tan^2 x}$$

Let


 * $$\tan \alpha = \frac{1}{5}$$

We can obtain tan(2&alpha;) = 5/12 and tan(4&alpha;) = 120/119 by using the above formula. Therefore,


 * $$\tan ( 4 \tan^{-1} \frac{1}{5} ) = \frac{120}{119}$$

Consider,


 * $$\tan (4 \tan^{-1} \frac{1}{5} - \tan^{-1} \frac{1}{239}) = \frac{\frac{120}{119} - \frac{1}{239}}{1 + \frac{120}{119} \frac{1}{239}} = \frac{120 \cdot 239 - 119}{119 \cdot 239 + 120} = \frac{120(120 + 119) - 119}{119(120 + 119) + 120} = \frac{120^2 + 119^2}{120^2 + 119^2} = 1$$


 * $$4 \tan^{-1} \frac{1}{5} - \tan^{-1} \frac{1}{239} = \frac{\pi}{4}$$

Q.E.D.