Question

докажите тождество sinx+cosx=√2cos(п/4-х)

Answer 1

$sinx+cosx = \sqrt{2}(cos \frac{ \pi }{4}-x)$
$\sqrt{2}cos( \frac{ \pi }{4}-x) = \sqrt{2}(cos \frac{ \pi }{4}*cosx+sin \frac{ \pi }{4}*sinx) =$ $\sqrt{2}( \frac{ \sqrt{2} }{2}*cosx+ \frac{ \sqrt{2} }{2}sinx) = \sqrt{2}* \frac{ \sqrt{2} }{2}(sinx+cosx) = \frac{2}{2}(sinx + cos x)$ $=sin x+ cos x$