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How many solutions does this equation have

$$2 \cos^2\left(\frac12 x \right) \sin^2 x = x^2+x-2$$

where $0 \lt x \le \displaystyle\frac \pi9?$

I observed that $2 \cos^2\left(\frac12x\right)$ can be written as $1+\cos x$. Simplifying $\sin^2 x,$ we get

$$(1+\cos x)^2(1-\cos x)=x^2+x-2$$

But I don't understand what to do after that.

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1 Answer 1

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HINT:

In the given range, the left hand side is positive. How about the right side?

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  • $\begingroup$ It is positive when x is greater than 1. $\endgroup$
    – Tejas
    Commented Nov 25, 2013 at 18:34
  • $\begingroup$ But that isn't in the given range. My concern is the sign of the right side in the given range. $\endgroup$ Commented Nov 25, 2013 at 18:37
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    $\begingroup$ Oh. $\frac \pi9$ is less than 1. So there cannot be any real solution. $\endgroup$
    – Tejas
    Commented Nov 25, 2013 at 18:39
  • $\begingroup$ Yeah! Done :D $ $ $\endgroup$ Commented Nov 25, 2013 at 18:40

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