Question

Find the interval in which $\left|\begin{array}{ccc}cosx& sinx& 1\\ sinx& cosx& 1\\ cos(x+y)& sin(x-y)& 0\end{array}\right|$ lies.

Answer 1

$\left|\begin{array}{ccc}cosx& sinx& 1\\ sinx& cosx& 1\\ cos(x+y)& sin(x-y)& 0\end{array}\right|[sinx.sin(x-y)-cos.cos(x+y)-[cosx.sin(x-y)-sinx.cos(x+y)]=sin(x-y\left)\right[sinx-cosx]+cos(x+y\left)\right[-cosx+sinx]=(sinx-cosx\left)\right[sin(x-y)+cos(x+y)\left]\right(sinx-cosx\left)\right[sinx.dy-cosx.siny+cosx.cosy-sinx.siny\left]\right(sinx-cosx\left)\right(cosy-siny\left)\right(sinx+cosx)=si{n}^2-co{s}^{2}x(cosy-siny)$
$-cos2x(cosy-siny)\sqrt{2}cos2x\left(\underset{\sqrt{2}}{1siny}\underset{\sqrt{2}}{-1cosy}\right)=\sqrt{2}cos2x.cos(y+\frac{\pi }{4})[-\sqrt{2},\sqrt{2}]$