Trigonometric Identities the Euler Way

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    "If we add (1) and (2), we can derive the cosine function:\n",
    "\n",
    "$$\\label{eq:3}\n",
    "\\cos\\theta = \\frac{e^{i\\theta} + e^{-i\\theta}}{2}. \\tag{3}\n",
    "$$\n",
    "\n",
    "If we subtract (2) from (1), we can derive the sine function:\n",
    "\n",
    "$$\\label{eq:4}\n",
    "\\sin\\theta = \\frac{e^{i\\theta} - e^{-i\\theta}}{2i}. \\tag{4}\n",
    "$$\n",
    "\n",
    "\n",
    "# Angle Addition Formulas\n",
    "\n",
    "Let's say we want to derive the angle addition formulas for sine and cosine using Euler's formula. We start with\n",
    "\n",
    "$$\n",
    "e^{i(\\alpha + \\beta)}  = e^{i\\alpha + i\\beta} = e^{i\\alpha} e^{i\\beta}.\n",
    "$$\n",
    "\n",
    "Applying the Euler's formula to the left side, we have:\n",
    "\n",
    "$$\n",
    "e^{i(\\alpha + \\beta)} = \\cos(\\alpha + \\beta) + i \\sin(\\alpha + \\beta).\n",
    "$$\n",
    "\n",
    "Now, applying the Euler's formula to the right side, we have:\n",
    "\n",
    "$$\n",
    "e^{i\\alpha} e^{i\\beta} = (\\cos\\alpha + i \\sin\\alpha)(\\cos\\beta + i \\sin\\beta) =\n",
    "$$\n",
    "\n",
    "$$\n",
    "= \\cos\\alpha \\cos\\beta - \\sin\\alpha \\sin\\beta + i (\\sin\\alpha \\cos\\beta + \\cos\\alpha \\sin\\beta).\n",
    "$$\n",
    "\n",
    "Equating the real parts of the above two expressions, we get the **cosine addition formula**\n",
    "\n",
    "$$\n",
    "\\cos(\\alpha + \\beta) = \\cos\\alpha \\cos\\beta - \\sin\\alpha \\sin\\beta.\n",
    "$$\n",
    "\n",
    "Equating the imaginary parts, we get the **sine addition formula**\n",
    "\n",
    "$$\n",
    "\\sin(\\alpha + \\beta) = \\sin\\alpha \\cos\\beta + \\cos\\alpha \\sin\\beta.\n",
    "$$"
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