Question

решить уравнение
1)√2cos(pi/4 + x) - cosx=1
2)√2sin(pi/4 - x/2) + sin(x/2)=1

Answer 1

$sqrt{2} cos( frac{ pi }{4} +x)-cosx=1 sqrt{2} (cos frac{ pi }{4}cosx-sin frac{ pi }{4}sinx )-cosx=1$
$sqrt{2} ( frac{ sqrt{2} }{2}cosx- frac{ sqrt{2} }{2} sinx )-cosx=1$$cosx-sinx-cosx=1$
-sinx=1
$x=- frac{ pi }{2} =2 pi n$

$sqrt{2} sin( frac{ pi }{4}- frac{x}{2} )+sin frac{x}{2} =1$$sqrt{2} (sin frac{ pi }{4}cos frac{x}{2} -cos frac{ pi }{4} sin frac{x}{2} +sin frac{x}{2} )=1$$cos frac{x}{2} -sin frac{x}{2} +sin frac{x}{2} =1$$cos frac{x}{2} =1 frac{x}{2} =2 pi k x=4 pi k$