The general solution of cot frac{x}{2} - cot | vedclass

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(B) Given equation: $\cot \frac{x}{2} - \cot x = \operatorname{cosec} \frac{x}{2}$Using $\cot 	heta = \frac{\cos 	heta}{\sin 	heta}$ and $\operator

Vedclass · Accepted Answer

$\{4n\pi \pm \frac{2\pi}{3} \mid n \in Z\}$

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(B) Given equation: $\cot \frac{x}{2} - \cot x = \operatorname{cosec} \frac{x}{2}$Using $\cot 	heta = \frac{\cos 	heta}{\sin 	heta}$ and $\operatorname{cosec} 	heta = \frac{1}{\sin 	heta}$:$\frac{\cos(x/2)}{\sin(x/2)} - \frac{\cos x}{\sin x} = \frac{1}{\sin(x/2)}$Multiply by $\sin(x/2)$ (assuming $\sin(x/2) 
eq 0$):$\cos(x/2) - \frac{\cos x \cdot \sin(x/2)}{\sin x} = 1$$\cos(x/2) - \frac{\cos x \cdot \sin(x/2)}{2 \sin(x/2) \cos(x/2)} = 1$$\cos(x/2) - \frac{\cos x}{2 \cos(x/2)} = 1$$2 \cos^2(x/2) - \cos x = 2 \cos(x/2)$Since $2 \cos^2(x/2) = 1 + \cos x$:$1 + \cos x - \cos x = 2 \cos(x/2)$$1 = 2 \cos(x/2) \Rightarrow \cos(x/2) = \frac{1}{2}$$\cos(x/2) = \cos(\frac{\pi}{3})$$\frac{x}{2} = 2n\pi \pm \frac{\pi}{3}$$x = 4n\pi \pm \frac{2\pi}{3}, n \in Z$

The general solution of $\cot \frac{x}{2} - \cot x = \operatorname{cosec} \frac{x}{2}$ is

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