<s>[INST] If $\sin x + \csc x =2 \tan x$. Find value of $\cos^9x +\cot^9x +\sin^7x$ Problem: 

If $\sin x+\csc x=2\tan x$, Find value of $\cos^9x+\cot^9x+\sin^7x$

Solution: 
\begin{align*}&\sin x+\csc x=2\tan x \\ &\sin x+\frac{1}{\sin x}=2\frac{\sin x}{\cos x} \\ &\sin^2x+1=2\frac{\sin^2x}{\cos x} \\ &\sin^2x\cos x+\cos x=2\sin^2x \\ &(1-\cos^2x)\cos x+\cos x=2(1-\cos^2x) \\ &\cos^3x-2\cos^2x-2\cos x+2=0\end{align*}
Am I doing right ?
How to do further ?
 [/INST] This is not really an answer, but a comment with image indicating that something is wrong with the problem in case the options are $1$, $0$, $-1$ and $2$ as stated in a comment above.
In the picture below I have let Mathematica draw the graphs of $\sin x+\csc x$ (blue), $2\tan x$ (yellow), $\cos^9x+\cot^9x+\sin^7x$ (green), $1$ (red) and $-1$ (purple). 
It should be clear that, when the blue and yellow graphs meet each other, the green is not at any of the values stated in the problem.

To make it more evident, here is a zoomed graph around the first of the two points where the graphs meet:

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