Question

FIND...................

Answer 1

$\cos ( \dfrac{ x }{ 2 } ) \cos ( \dfrac{ x }{ { 2 }^{ 2 } } ) \sin ( \dfrac{ x }{ { 2 }^{ 2 } } )$$= \frac {sin(x)}{4}$

$\mathrm{Use\:the\:following\:identity}:\quad \cos \left(x\right)\sin \left(x\right)=\frac{\sin \left(2x\right)}{2}$

$=\frac{\sin \left(2\cdot \frac{x}{2^2}\right)}{2}\cos \left(\frac{x}{2}\right)$

$\mathrm{Multiply\:fractions}:\quad \:a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c}$

$=\frac{\sin \left(2\cdot \frac{x}{2^2}\right)\cos \left(\frac{x}{2}\right)}{2}$

$\mathrm{Multiply\:}2\cdot \frac{x}{2^2}\::\quad \frac{x}{2}$

$=\frac{\sin \left(\frac{x}{2}\right)\cos \left(\frac{x}{2}\right)}{2}$

$\mathrm{Simplify}\:\sin \left(\frac{x}{2}\right)\cos \left(\frac{x}{2}\right):\quad \frac{\sin \left(2\cdot \frac{x}{2}\right)}{2}$

$=\frac{\frac{\sin \left(2\cdot \frac{x}{2}\right)}{2}}{2}$

$\frac{\sin \left(2\cdot \frac{x}{2}\right)}{2}=\frac{\sin \left(x\right)}{2}$

$=\frac{\frac{\sin \left(x\right)}{2}}{2}$

$\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{\frac{b}{c}}{a}=\frac{b}{c\:\cdot \:a}$

$=\frac{\sin \left(x\right)}{2\cdot \:2}$

$\mathrm{Multiply\:the\:numbers:}\:2\cdot \:2=4$

$=\frac{\sin \left(x\right)}{4}$