Machin's Formula for Pi/Proof 2

Proof
$$\begin{align} \arg(a+bi) &= \arctan(b/a) \ \text{if} \ \ b > 0 \\ \arg(X \cdot Y) &= \arg(X) + \arg (Y)\\\ \\ \arg((5+i)^4\cdot(239-i)) &= \arg((5+i)^4) + \arg(239-i) \\ \arg((476+480i)\cdot(239-i)) &= 4\arg(5+i) \ \ + \arg(239-i) \\ \arg(114244(1+i)) &= 4\arctan(1/5) + \arctan(-1/239) \\ \frac{\pi}{4} &= 4\arctan(1/5) - \arctan(1/239) \\ \end{align}$$

$$\begin{align} (5+i)^4 &= 5^4 + 4 \cdot 5^3i - 6 \cdot 5^2 - 4 \cdot 5i + 1 \\ &= 625 + 500i - 150 - 20i + 1 \\ &= 476 + 480i \\\ \\ (5+i)^4 \cdot (-239+i) &= (476+480i) \cdot (-239+i) \\ &= -113764-480-114720i+476i \\ &= -114244 - 114244i \\ &= -114244(1+i) \\ \end{align}$$