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Question Number 43877 by peter frank last updated on 16/Sep/18

$$\tan \frac{x}{2}=\mathrm{cosec}-\sin xprovethat$$$$\tan {}^2\frac{x}{2}=(-2\pm \sqrt{5})$$

Answered by ajfour last updated on 16/Sep/18

$$\tan \frac{x}{2}=\frac{1}{\sin x}-\sin x$$$$let\tan \frac{x}{2}=t\Rightarrow \sin x=\frac{2t}{1+t^2}$$$$\Rightarrow t=\frac{1+t^2}{2t}-\frac{2t}{1+t^2}$$$$\Rightarrow \frac{1}{2t}-\frac{t}{2}=\frac{2t}{1+t^2}$$$$\Rightarrow 1-t^4=4t^2$$$$or{(t^2+2)}^2=5$$$$t^2=\tan {}^2\frac{x}{2}=-2+\sqrt{5}.$$

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