sin x + cos x + 2.sqrt(2) . sin x . cos x= 0sinx+cosx+2.√2.sinx.cosx=0 or
cosx = -sinx/(1+2sqrt2 sinx)cosx=−sinx1+2√2sinx
now making y = sinxy=sinx we have
sqrt(1-y^2) = -y/(1+2sqrt2 y)^2√1−y2=−y(1+2√2y)2 or squaring both sides
1-y^2=y^2/(1+2sqrt2y)^21−y2=y2(1+2√2y)2 or
(1-y^2)(1+2sqrt2y)^2-y^2=0(1−y2)(1+2√2y)2−y2=0 or
(2y^2+2sqrt2y+1)(-4y^2+2sqrt2y+1)=0(2y2+2√2y+1)(−4y2+2√2y+1)=0 then we have
{(2y^2+2sqrt2y+1=0->{(y=-1/sqrt2),(y=-1/sqrt2):}),(-4y^2+2sqrt2y+1=0->{(y=1/4 (sqrt[2] - sqrt[6])),(y=1/4 (sqrt[2] + sqrt[6])):}):}
or
sinx={(-1/sqrt2->x = (5pi)/4 + 2 k pi),(1/4 (sqrt[2] - sqrt[6])->x = arcsin(1/4 (sqrt[2] - sqrt[6]))+2kpi),(1/4 (sqrt[2] + sqrt[6])->x = pi-arcsin(1/4 (sqrt[2] + sqrt[6]))+2kpi):}
