Question

Докажите равенство
$displaystyle frac{sin50^circ+cos50^circ}{sqrt{2}sin85^circ}=1;\ frac{cos40^circ-sqrt{3}sin40^circ}{sin190^circ}=2$

Answer 1

$1)quad frac{sin50+cos50}{sqrt2sin95} = [, cos50=cos(90-40)=sin40]=\\=frac{sin50+sin40}{sqrt2sin(90-5)} = frac{2sin45cdot cos5}{sqrt2cos5} = frac{2cdot frac{sqrt2}{2}}{sqrt2} = frac{2sqrt2}{2sqrt2} =1\\2)quad frac{cos40-sqrt3sin40}{sin190} = frac{2(frac{1}{2}cos40-frac{sqrt3}{2}sin40)}{sin(180+10)} = frac{2(sin30cdot cos40-cos30cdot sin40)}{-sin10} =\\= frac{2cdot sin(30-40)}{-sin10} = -frac{2sin(-10)}{-sin10} = frac{2sin10}{sin10} =2$