Originally Posted by

**patzer** Hello!

(**First problem) Express $\displaystyle sin5x$ in terms of $\displaystyle a) sinx$ and $\displaystyle b) cosx$**

**Attempt:** According to **de Moivre's formula** - $\displaystyle (cosx+isinx)^n=cos(nx)+isin(nx)$ so:

**de Moivre** - $\displaystyle (cosx+isinx)^5=cos(5x)+isin(5x)$ and

**binomial** - $\displaystyle (cosx+isinx)^5=cos^5(x)+5cos^4(x)isin(x)-10cos^3(x)sin^2(x)-10cos^2(x)isin^3(x)+5cos(x)sin^4(x)+sin^5(x)i$

So,

(1)$\displaystyle sin5x=5cos^4(x)sin(x)-10cos^2(x)sin^3(x)+sin^5(x)$

After some arithmetic arrangements and and **Pythagorean trigonometric identity** I got this: $\displaystyle sin5x=12sin^5(x)-12sin^3(x)+sin(x)$ and task under a) is completed (I think).

But the problem is at task b), so, I don't know how to write expression (1) in terms of cosx function. Is there any identity? Or maybe some other method.

(**Second problem) If $\displaystyle z$ is a complex number, what does this equation represent: $\displaystyle |z|-2Re(z)=4$**

**Attempt:** No attempt. I need a complete solution and explanation.

Thank you!