# How do i proof tan2x=(2tanx)/(1-tan^2x) with moivre?

Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

1
Feb 9, 2018

By de Moivre's theorem we can write

$\cos 2 x + i \sin 2 x = {\left(\cos x + i \sin x\right)}^{2}$

$\implies \cos 2 x + i \sin 2 x = {\cos}^{2} x - {\sin}^{2} x + i \cdot 2 \sin x \cos x$

So equating real parts of both sides we get
$\cos 2 x = {\cos}^{2} x - {\sin}^{2} x$

And equating imaginary parts of both sides we get
$\sin 2 x = 2 \sin x \cos x$

Hence

$\tan 2 x = \frac{\sin 2 x}{\cos 2 x} = \frac{2 \sin x \cos x}{{\cos}^{2} x - {\sin}^{2} x}$

$= \frac{\frac{2 \sin x \cos x}{\cos} ^ 2 x}{{\cos}^{2} \frac{x}{\cos} ^ 2 x - {\sin}^{2} \frac{x}{\cos} ^ 2 x}$

$= \frac{2 \tan x}{1 - {\tan}^{2} x}$

• 10 minutes ago
• 10 minutes ago
• 11 minutes ago
• 12 minutes ago
• A minute ago
• A minute ago
• 2 minutes ago
• 5 minutes ago
• 6 minutes ago
• 10 minutes ago
• 10 minutes ago
• 10 minutes ago
• 11 minutes ago
• 12 minutes ago