Question

Prove that
$\large \tt \red{{e}^{i \theta} = \cos \theta + i \sin \theta}$​

Answer 1

Answer:

$we \: know \: that \: {e}^{x} = 1 + \frac{x}{1!} + \frac{ {x}^{2} }{2!} + \frac{ {x}^{3} }{3!} + ... \\ similarly \: {e}^{ix} = 1 + \frac{ix}{1!} + \frac{ {ix}^{2} }{2!} + \frac{ {ix}^{3} }{3!} + ... \\ = 1 + \frac{ix}{1!} + \frac{ {i}^{2} {x}^{2} }{2!} + \frac{ {i}^{3} {x}^{3} }{3!} + ... \\ = 1 + \frac{ix}{1!} - \frac{ {x}^{2} }{2!} - \frac{ i{x}^{3} }{3!} + \frac{ {x}^{4} }{4!} + ... \\ = (1 - \frac{ {x}^{2} }{2!} + \frac{ {x}^{4} }{4!} - ...) +i (\frac{x}{1!} - \frac{ {x}^{3} }{3!} + \frac{ {x}^{5} }{5!} - ...) \\ = \cos(x) + i \sin(x)$

you can put theta incase of x