# 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 ?

• Where did this problem come from? You can obviously solve the cubic equation you have got for $\cos x$ numerically (it has only one real root in the range [-1,1]). Note that you will get two possible values for $\sin x$ and hence two different possible values for $\cos^9x+\cot^9x+\sin^7x$. But you may be expected to do some clever manipulation. It depends on the source of the problem. – almagest Jun 14 '16 at 6:25
• Its an objective question. Options are a) $1$ b) $0$ c) $-1$ and d) $2$ – rst Jun 14 '16 at 6:47
• I don't think any of those options are correct. Are you sure that the statement of the problem is correctly written? – mickep Jun 14 '16 at 6:55
• yeah, statement of question is correct. – rst Jun 14 '16 at 7:12

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).