Question

if 13 theta = pi show that cos 3 theta + cos 5 theta + 2 cos theta cos 9 theta = zero

Answer 1

$\textbf{Given:}$

$13\,\theta=\pi$

$\textbf{To prove:}$

$cos3\theta+cos5\theta+2\;cos\theta\;cos9\theta=0$

$\text{Consider,}$

$cos3\theta+cos5\theta+2\;cos\theta\;cos9\theta$

$\text{Using}$

$\boxed{\bf\;cosC+cosD=2\;cos(\dfrac{C+D}{2})\;cos(\dfrac{C-D}{2})}$

$=2\;cos(\dfrac{3\theta+5\theta}{2})\;cos(\dfrac{3\theta-5\theta}{2})+2\;cos\theta\;cos9\theta$

$=2\;cos4\theta\;cos(-\theta)+2\;cos\theta\;cos9\theta$

$=2\;cos4\theta\;cos\theta+2\;cos\theta\;cos9\theta$

$=2\;cos\theta[cos4\theta+cos9\theta]$

$=2\;cos\theta[2\;cos(\dfrac{4\theta+9\theta}{2})\;cos(\dfrac{4\theta-9\theta}{2})]$

$=2\;cos\theta[2\;cos\dfrac{13\theta}{2}\;cos(\dfrac{-5\theta}{2})]$

$=4\;cos\theta\;cos\dfrac{\pi}{2}\;cos\dfrac{5\theta}{2}$

$=4\;cos\theta(0)cos\dfrac{5\theta}{2}$

$=0$

$\therefore\boxed{\bf\;cos3\theta+cos5\theta+2\;cos\theta\;cos9\theta=0}$

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