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

[tex]sinx+cosx = \sqrt{2}(cos \frac{ \pi }{4}-x) \\ [/tex]
[tex] \sqrt{2}cos( \frac{ \pi }{4}-x) = \sqrt{2}(cos \frac{ \pi }{4}*cosx+sin \frac{ \pi }{4}*sinx) = [/tex] [tex]\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) [/tex] [tex]=sin x+ cos x[/tex]