If $\sin x+\cos ecx=-2$, then for a positive integer $n$, ${\sin }^{n}x+\cos e{c}^{n}x$ is

$\sin x+\cos ecx=-2$

Squaring on both sides

$\Rightarrow {\sin }^{2}x\text{\hspace{0.17em}}+\cos e{c}^{2}x+2=4$

$\Rightarrow {\sin }^{2}x+\cos e{c}^{2}x=2$(When $n$ is even)

$\sin x+\cos ecx=-2$

Cubing on both sides

$\Rightarrow {\sin }^{3}x+\cos e{c}^{3}x+3\times 1\times (\sin x+\cos ecx)=-8$

$\Rightarrow {\sin }^3+\cos e{c}^{3}x-6=-8$

$\Rightarrow {\sin }^3+\cos e{c}^{3}x=-8+6$

$=-2$

$nisodd$

$\sin x+\cos ecx=-2$