Question

Prove that :-



$\quad \qquad { \bigstar { \boxed { \underline { \underline { \boxed { \red { \bf { {e}^{ix} = Cos x + i.Sin x }}}}}}}}{\bigstar}$ ​

Answer 1

We know that :

$\\ \cos \text x = 1 - \frac{ {\text x}^{2} }{2!} + \frac{ {\text x}^{4} }{4!} - \: . \: . \: . \: .$

$\sin \text x = {\text x} - \frac{ {\text x}^{3} }{3!} + \frac{ {\text x}^{5} }{5!} - . \: .\: . \:.$

Also we know ;

$\rm {e}^{x} = 1 + x + \frac{ {x}^{2} }{2!} + \frac{ {x}^{3} }{3!} + \frac{ {x}^{4} }{4!} + \frac{ {x}^{5} }{5!} + \: . \: . \: . \: .$

$\dashrightarrow \text{Replacing \: x \: by} \: i \text x$

$: \implies {\text e}^{i \text x} = 1+ \frac{i\text x}{1!} + \frac{ {i}^{2} {\text x}^{2} }{2!} + \frac{ {i}^{3} {\text x}^{3} }{3!} + \frac{ {i}^{4} {\text x}^{4} }{4!} + \frac{ {i}^{5} {\text x}^{5} }{5!} + . \: .\: . \: .\: .$

$= 1 + {i\text x} + \frac{ ( - 1){\text x}^{2} }{2!} + \frac{ ( - i){\text x}^{3} }{3!} + \frac{ (1) {\text x}^{4} }{4!} + \frac{ ({i}) {\text x}^{5} }{5!} + \: .\: .\: . \: .$

➡️ Separating imaginary and real terms ;

$\\ : \implies {\text e}^{i \text x} = \bigg[ 1 - \frac{ {\text x}^{2} }{2!} + \frac{ {\text x}^{4} }{4!} - \: . \: . \: . \: . \bigg] + i \bigg[{ {\text x} } - \frac{ {\text x}^{3} }{3!} + \frac{ {\text x}^{5} }{5!} - . \: .\: . \:. \bigg]$

➡️ Using value of cos(x) and sin(x) :

$\purple{ \large : \implies \underline {\boxed{ \pmb{ {\text e}^{i \text x} = \cos \text x + i \sin \text x} }}}$

$\large \blue \maltese \: \: \underline{\pink{\underline{\frak{\pmb{\text Hence\:\:Proved }}}}}$

• The about proved formula is known as Euler's formula.

Answer 2

We know that :

$\\ \cos \text x = 1 - \frac{ {\text x}^{2} }{2!} + \frac{ {\text x}^{4} }{4!} - \: . \: . \: . \: .$

$\sin \theta = {\text x} - \frac{ {\text x}^{3} }{3!} + \frac{ {\text x}^{5} }{5!} - . \: .\: . \:.$

Also we know ;

$\rm {e}^{x} = 1 + x + \frac{ {x}^{2} }{2!} + \frac{ {x}^{3} }{3!} + \frac{ {x}^{4} }{4!} + \frac{ {x}^{5} }{5!} + \: . \: . \: . \: .$

$\dashrightarrow \text{Replacing \: x \: by} \: i \text x$

$: \implies {\text e}^{i \text x} = 1+ \frac{i\text x}{1!} + \frac{ {i}^{2} {\text x}^{2} }{2!} + \frac{ {i}^{3} {\text x}^{3} }{3!} + \frac{ {i}^{4} {\text x}^{4} }{4!} + \frac{ {i}^{5} {\text x}^{5} }{5!} + . \: .\: . \: .\: .$

$= 1 + {i\text x} + \frac{ ( - 1){\text x}^{2} }{2!} + \frac{ ( - i){\text x}^{3} }{3!} + \frac{ (1) {\text x}^{4} }{4!} + \frac{ ({i}) {\text x}^{5} }{5!} + \: .\: .\: . \: .$

➡️ Separating imaginary and real terms ;

$\\ : \implies {\text e}^{i \text x} = \bigg[ 1 - \frac{ {\text x}^{2} }{2!} + \frac{ {\text x}^{4} }{4!} - \: . \: . \: . \: . \bigg] + i \bigg[{ {\text x} } - \frac{ {\text x}^{3} }{3!} + \frac{ {\text x}^{5} }{5!} - . \: .\: . \:. \bigg]$

➡️ Using value of cos(x) and sin(x) :

$\purple{ \large : \implies \underline {\boxed{ \pmb{ {e}^{i \text x} = \cos \text x + i \sin \text x} }}}$

$\large \blue \maltese \: \: \underline{\pink{\underline{\frak{\pmb{\text Hence\:\:Proved }}}}}$