Using Euler's formula for proving trigonometric identities

[Win_odd Dhamnekar]

With the aid of Euler's formula how to prove

 

[Dr.Peterson]

With the aid of Euler's formula how to prove

View attachment 30028

Have you tried expanding this? [math]\left(e^{ix}\right)^3 = \left(\cos(x)+i\sin(x)\right)^3[/math]