Trigonometric Identities the Euler Way

1.ipynb

2.ipynb

3.ipynb

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    "Let's derive the identity for the product of cosine and cosine functions. Using (3) and (4), we have\n",
    "\n",
    "$$\n",
    "\\cos\\alpha \\cos\\beta = \\frac{1}{4}\\left(e^{i\\alpha} + e^{-i\\alpha}\\right)\\left(e^{i\\beta} + e^{-i\\beta}\\right) =\n",
    "$$\n",
    "\n",
    "$$\n",
    "= \\frac{1}{4}\\left[e^{i(\\alpha + \\beta)} + e^{i(\\alpha - \\beta)} + e^{-i(\\alpha - \\beta)} + e^{-i(\\alpha + \\beta)}\\right] =\n",
    "$$\n",
    "\n",
    "$$\n",
    "= \\frac{1}{2}\\left[\\cos(\\alpha + \\beta) + \\cos(\\alpha - \\beta)\\right].\n",
    "$$\n",
    "\n",
    "# Conclusion\n",
    "\n",
    "The key advantage of using Euler's formula is that it transforms trigonometric problems into algebraic manipulations with exponentials, and I find this beautiful."
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