Could anyone help me with this problem that I've been struggling with?
Find all x such that:
$\displaystyle cos x + sin x = 1 + sin 2x$, for 0 \< x \< 2x where \< is greater than or equal to.
Thanks in advance.
Hello,
Please note that $\displaystyle (\sin(x)+\cos(x))^2=\sin^2(x)+\cos^2(x)+2 \sin(x) \cos(x)=1+\sin(2x)$
So your equation is :
$\displaystyle (\cos(x)+\sin(x))^2=(1+\sin(2x))^2$
$\displaystyle 1+\sin(2x)=(1+\sin(2x))^2$
$\displaystyle (1+\sin(2x))(1+\sin(2x)-1)=0$
$\displaystyle \sin(2x) \cdot (1+\sin(2x))=0$