MHB Trig identities Fourier Analysis

[Dustinsfl]

Prove the identities
$$
\frac{\sin\left(\frac{n + 1}{2}\theta\right)}{\sin\frac{\theta}{2}}\cos\frac{n}{2}\theta = \frac{1}{2} + \frac{\sin\left(n + \frac{1}{2}\right)\theta}{2\sin\frac{\theta}{2}}
$$
By using the identity $\sin\alpha + beta$, I was able to obtain the $1/2$ but now I am not to sure with what to do.
$$
\frac{(\sin\frac{n\theta}{2}\cos\frac{\theta}{2}+\sin\frac{\theta}{2}\cos\frac{n\theta}{2})\cos\frac{n\theta}{2}}{\sin\frac{\theta}{2}} = \frac{1}{2} + \frac{1}{2}\cos n\theta + \frac{\sin\frac{n\theta}{2}\cos\frac{n\theta}{2} \cos\frac{\theta}{2}}{\sin\frac{\theta}{2}}
$$

Any suggestions?

 

[Amer]

\frac{(\sin\frac{n\theta}{2}\cos\frac{\theta}{2}+\ sin\frac{\theta}{2}\cos\frac{n\theta}{2})\cos\frac {n\theta}{2}}{\sin\frac{\theta}{2}} = \frac{1}{2} + \frac{1}{2}\cos n\theta + \frac{\sin\frac{n\theta}{2}\cos\frac{n\theta}{2} \cos\frac{\theta}{2}}{\sin\frac{\theta}{2}}

\frac{1}{2} + \frac{1}{2}\cos n\theta + \frac{\sin\frac{n\theta}{2}\cos\frac{n\theta}{2} \cos\frac{\theta}{2}}{\sin\frac{\theta}{2}}

\frac{1}{2} + \frac{\cos n\theta \sin \frac{\theta}{2} + 2 \sin \frac{n\theta}{2} \cos \frac{n\theta}{2} \cos \frac{\theta}{2} }{ 2 \sin \frac{\theta}{2} }

using \sin 2x = 2 \sin x \cos x

\frac{1}{2} + \frac{\cos n\theta \sin \frac{\theta}{2} + \sin n\theta \cos \frac{\theta}{2} }{ 2 \sin \frac{\theta}{2} }

note that \sin a+b = \sin a \cos b + \cos a \sin b

 

[Sudharaka]

Prove the identities
$$
\frac{\sin\left(\frac{n + 1}{2}\theta\right)}{\sin\frac{\theta}{2}}\cos\frac{n}{2}\theta = \frac{1}{2} + \frac{\sin\left(n + \frac{1}{2}\right)\theta}{2\sin\frac{\theta}{2}}
$$
By using the identity $\sin\alpha + beta$, I was able to obtain the $1/2$ but now I am not to sure with what to do.
$$
\frac{(\sin\frac{n\theta}{2}\cos\frac{\theta}{2}+\sin\frac{\theta}{2}\cos\frac{n\theta}{2})\cos\frac{n\theta}{2}}{\sin\frac{\theta}{2}} = \frac{1}{2} + \frac{1}{2}\cos n\theta + \frac{\sin\frac{n\theta}{2}\cos\frac{n\theta}{2} \cos\frac{\theta}{2}}{\sin\frac{\theta}{2}}
$$

Any suggestions?

Hi dwsmith, :)

This can be proved using the Product to Sum identity.

\begin{eqnarray}

\frac{\sin\left(\frac{n + 1}{2}\theta\right) \cos\frac{n}{2}\theta}{\sin\frac{\theta}{2}}&=& \frac{\sin\left( \frac{2n + 1}{2}\theta\right)+\sin \left(\frac{\theta}{2} \right)}{2\sin\frac{\theta}{2}}\\

&=& \frac{1}{2} + \frac{\sin\left(n + \frac{1}{2}\right)\theta}{2\sin\frac{\theta}{2}}

\end{eqnarray}

Kind Regards,
Sudharaka.