Unizor - Trigonometry and Complex Numbers - Problems 1

Using the Euler's formula
e^(i·x) = cos(x)+i·sin(x),
prove the following equalities for any real numbers x and y.

1. e^(i·0) = 1

2. e^(i·x)·e^(i·y) = e^[i·(x+y)]

3. 1/[e^(i·x)]= e^(−i·x)

4. [e^(i·x)]^y = e^(i·x·y)

5. Absolute value (modulus) of e^(i·x) equals to 1

6. Multiplication of any complex number represented on a coordinate plane by e^(i·x) results in its rotation by an angle of x radians.