# How to show that? A(x)=4cosx.cos2x.sin4x

## For each x of R we put: A(x): Sinx+Sin3x+Sin5x+Sin7x

Feb 10, 2018

#### Explanation:

We use here the identity relations -

$\sin A + \sin B = 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)$ and

$\cos A + \cos B = 2 \cos \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)$

$\sin x + \sin 3 x + \sin 5 x + \sin 7 x$

= $\sin 7 x + \sin x + \sin 5 x + \sin 3 x$

= $2 \sin \left(\frac{7 x + x}{2}\right) \cos \left(\frac{7 x - x}{2}\right) + 2 \sin \left(\frac{5 x + 3 x}{2}\right) \cos \left(\frac{5 x - 3 x}{2}\right)$

= $2 \sin 4 x \cos 3 x + 2 \sin 4 x \cos x$

= $2 \sin 4 x \left[\cos 3 x + \cos x\right]$

= $2 \sin 4 x \times 2 \cos \left(\frac{3 x + x}{2}\right) \cos \left(\frac{3 x - x}{2}\right)$

= $4 \sin 4 x \cos 2 x \cos x$