Question

напишите в мультипликативной форме: 1+sinx+cosx​

Answer 1

Ответ:

$\displaystyle1+sin x+cos x=2\sqrt 2cos \frac{x}{2}cos(\frac{\pi}{4}-\frac{x}{2})$

Объяснение:

$\displaystyle1+sin x+cos x=1+sin (2\cdot \frac{x}{2})+cos(2\cdot \frac{x}{2})=\\\\\\1+2sin \frac{x}{2}cos \frac{x}{2}+2cos^2\frac{x}{2}-1=2sin \frac{x}{2}cos \frac{x}{2}+2cos^2\frac{x}{2}=\\\\\\2cos \frac{x}{2}(sin \frac{x}{2}+cos\frac{x}{2})=2cos \frac{x}{2}(cos(\frac{\pi}{2}- \frac{x}{2})+cos\frac{x}{2})=$

$\displaystyle 2cos \frac{x}{2}\cdot 2cos\frac{\frac{\pi}{2}- \frac{x}{2}+\frac{x}{2}}{2}cos\frac{\frac{\pi}{2}- \frac{x}{2}-\frac{x}{2}}{2}=\\\\\\ 4cos \frac{x}{2}\cdot cos\frac{\pi}{4}cos\frac{\frac{\pi}{2}-x}{2}=4cos \frac{x}{2}\cdot \frac{\sqrt2}{2}cos(\frac{\pi}{4}-\frac{x}{2})=\\\\\\ 2\sqrt 2cos \frac{x}{2}cos(\frac{\pi}{4}-\frac{x}{2})$