$1.tanx + cotx=-2\sqrt{2}$
$\Leftrightarrow \frac1{sinx.cosx}=-2\sqrt{2}$
$\Leftrightarrow sin2x=\frac{-1}{\sqrt{2}}$
$\Leftrightarrow x= -\frac{\pi}8 + k\pi$
hc $x=\frac{5\pi}8 + k\pi$
2.$sinx + cosx =\sqrt{2}$
$\Leftrightarrow \sqrt{2}sin(x+\frac{\pi}4)=\sqrt{2}$
3. $sin^6 + cos^6=\frac7{16}$
$sin^6 + cos^6 = (sin^2+cos^2)(sin^4 + cos^4 -sin^2.cos^2)=sin^4 + cos^4 - sin^2.cos^2$
$=(sin^2 + cos^2)^2 -3sin^2.cos^2=1-\frac3{4}sin^2(2x)=\frac5{8} - cos4x\Rightarrow cos 4x=\frac3{16}$