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Prove $$\dfrac{2\sin x+\sin 2x}{2\sin x-\sin 2x}=\csc^2x+2\csc x \cot x+\cot^2x$$

Proving right hand side to left hand side:

$$\begin{align}\csc^2x+2\csc x \cot x+\cot^2x &= \frac{1}{\sin^2x}+\dfrac{2\cos x}{\sin^2x}+\dfrac{\cos^2x}{\sin^2x} \tag1 \\[0.8em] &=\frac{\cos^2x+2\cos x+1}{\sin^2x} \tag2 \\[0.8em] &=\frac{1-\sin^2x+1+\dfrac{\sin 2x}{\sin x}}{\sin^2x} \tag3 \\[0.8em] &=\frac{\;\dfrac{2\sin x-\sin^3x+\sin 2x}{\sin x}\;}{\sin^2x} \tag4 \\[0.8em] &=\frac{2\sin x-\sin^3x+\sin2 x}{\sin^3x} \tag5 \end{align}$$

I could not prove further to the left hand side from here. I would need help. Thank you in advance.

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    $\begingroup$ Hint : $\sin2x=2\sin x\cos x$ $\endgroup$
    – Tryhard
    Sep 25, 2021 at 10:00
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    $\begingroup$ You can obtain $\sin x$, $\cos x$, $\tan x$, $\csc x$, $\sec x$, and $\cot x$ by typing $\sin x$, $\cos x$, $\tan x$, $\csc x$, $\sec x$, and $\cot x$, respectively. $\endgroup$ Sep 25, 2021 at 10:04

3 Answers 3

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We have that, using $\sin 2x=2\sin x\cos x$ and cancelling out $2\sin x\neq0$

$$\dfrac{2\sin x+\sin 2x}{2\sin x-\sin 2x}= \dfrac{2\sin x+2\sin x\cos x}{2\sin x-2\sin x\cos x}= \dfrac{1+\cos x}{1-\cos x}$$

then, assuming $\cos x \neq -1$ ( which holds since we need $\sin x\neq0$ for the original expression), multiply by $\frac{1+\cos x} {1+\cos x}$ and simplify to obtain the result using that $1-\cos^2x=\sin^2x$.

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The following relations would be used in the answer:$$1-\cos^2x = \sin^2x$$ $$\sin 2x=2\sin x\cos x$$
Taking LHS, $$\frac{2\sin x+\sin 2x}{2\sin x-\sin 2x} = \frac{2\sin x(1+\cos x)}{2\sin x(1-\cos x)}$$ Cancelling $\sin 2x$

$$=\frac{1+\cos x}{1-\cos x}$$ Multiplying $1+\cos x$ to numerator and denominator $$=\frac{(1+\cos x)^2}{1-\cos^2x}$$ which is equivalent to $$=\frac{(1+\cos^2x+2\cos x)}{\sin^2x}$$ Divinding each term by denominator, we get LHS as $$=\csc^2x+2\csc x\cot x+\cot^2x$$ which is the RHS.

Hence, Proved.

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$$\dfrac{2\sin x+\sin 2x}{2\sin x-\sin 2x}=$$

$$ \frac {2\sin x +2\sin x \cos x }{2\sin x -2\sin x \cos x}=$$ $$\frac {1 + \cos x }{1 -\cos x}=$$

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

$$\frac { 1+2\cos x +\cos ^2 x}{\sin ^2 x}=$$

$$\csc^2x+2\csc x \cot x+\cot^2x$$

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