Question

Calculaţi valoarea expresiei .(sin^2 \pi /8- cos^2 \pi /8)^2

Answer 1

$\big(sin^2 \frac{\pi}{8} -cos^2 \frac{\pi}{8}\big)^2 = \Big[-\big(cos^2 \frac{\pi}{8} -sin^2 \frac{\pi}{8}\big)\Big]^2 = \big(-cos(2\cdot\frac{\pi}{8})\big) ^2 = \\ \\ =(-cos\frac{\pi}{4})^2 = \big(-\frac{ \sqrt{2} }2}\big)^2 = \dfrac{2}{4} = \dfrac{1}{2}$

Answer 2

$(sin^2 \frac{ \pi }{8}-cos^2 \frac{ \pi }{8})^2=(sin \frac{ \pi }{8}-cos \frac{ \pi }{8})^2(sin \frac{ \pi }{8}+cos \frac{ \pi }{8})^2= \\ =(sin^2 \frac{ \pi }{8}+cos^2 \frac{ \pi }{8}-2sin \frac{ \pi }{8}cos \frac{ \pi }{8})(sin^2 \frac{ \pi }{8}+cos^2 \frac{ \pi }{8}+2sin \frac{ \pi }{8}cos \frac{ \pi }{8})= \\ =(1-sin2 \frac{ \pi }{8})(1+sin2 \frac{ \pi }{8})=(1- sin\frac{ \pi }{4})(1+sin \frac{ \pi }{4})= \\ =(1-sin^2 \frac{ \pi }{4})=cos^2 \frac{ \pi }{4}=( \frac{ \sqrt{2} }{2})^2$
$= \frac{2}{4}= \frac{1}{2}$