$\sqrt{2A^2(1+\cos \theta)} $
to
$\sqrt{4A^2\cos^2 \dfrac {\theta} {2}}$ (Here, divided by $2$ is only under $\theta$)
I have solved till : $\dfrac {2A\sqrt{1+\cos\theta}}{2}$ (divided by $2$ is whole under $1 + \cos \theta$)
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$\begingroup$ $\cos(\theta)=\cos (2\theta/2)=2\cos^2(\theta/2)-1\implies 2A^2(1+\cos(\theta))=4A\cos^2(\theta/2) $ $\endgroup$– KoroMar 15, 2021 at 10:16
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$\begingroup$ @Koro Thanks. How did you solve the 3rd step ? $\endgroup$– RiderMar 15, 2021 at 10:18
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$\begingroup$ Your "till" result is wrong. $\endgroup$– user65203Mar 15, 2021 at 10:52
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$\begingroup$ @Koro: I think you're missing a factor of $A$ when you went to the third step (it should be $4A^2)$. $\endgroup$– bjcolby15Mar 15, 2021 at 11:21
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1$\begingroup$ @bjcolby15: Ooops, I missed that. You are right. It should be $4A^2$ instead of $4A$ in last step. $\endgroup$– KoroMar 15, 2021 at 11:29
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1 Answer
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Hint: Using the trigonometric identity (a variation of the half-angle formula) $$\cos^2 \dfrac {\theta}{2} = \dfrac {1 + \cos \theta}{2}$$ rewrite this expression in terms of $1 + \cos \theta$ only, and the answer will be immediate.
answered Mar 15, 2021 at 10:43