Question

Show that
a) cos x cos 7x - cos 3x cos 11x=
sin 4x sin 10x

Solution..... ​

Answer 1

Answer:

$cosx\:cos7x-cos3x\:cos11x=sin10x\:sin4x$

Step-by-step explanation:

Formula used:

$\boxed{cos(A+B)+cos(A-B)=2\:cosA\:cosB}$

$\boxed{cosC-cosD=-2\:sin(\frac{C+D}{2})\:sin(\frac{C-D}{2})}$

$cosx\:cos7x-cos3x\:cos11x$

$=\frac{1}{2}[2\:cosx\:cos7x-2\:cos3x\:cos11x]$

$=\frac{1}{2}[cos(x+7x)+cos(x-7x)-(cos(3x+11x)+cos(3x-11x))]$

$=\frac{1}{2}[cos8x+cos(-6x)-(cos14x+cos(-8x))]$

$=\frac{1}{2}[cos8x+cos6x-cos14x-cos8x]$

$=\frac{1}{2}[cos6x-cos14x]$

$=\frac{1}{2}[-2\:sin(\frac{6x+14x}{2})\:sin(\frac{6x-14x}{2})]$

$=\frac{1}{2}[-2\:sin10x\:sin(-4x)]$

$=\frac{1}{2}[2\:sin10x\:sin4x]$

$=sin10x\:sin4x$

$\implies\:\boxed{cosx\:cos7x-cos3x\:cos11x=sin10x\:sin4x}$