A) \[\tan \left( \frac{\alpha +\beta }{2} \right)\,\,\tan \left( \frac{\alpha -\beta }{2} \right)\]
B) \[\cot \left( \frac{\alpha +\beta }{2} \right)\,\,\cot \left( \frac{\alpha +\beta }{2} \right)\]
C) \[\cot \left( \frac{\alpha -\beta }{2} \right)\,\,\cot \left( \frac{\alpha +\beta }{2} \right)\]
D) \[\cot \left( \frac{\alpha +\beta }{2} \right)\,\,\cot \left( \frac{\alpha +\beta }{2} \right)\]
Correct Answer: C
Solution :
Given, \[{{z}_{1}}=\cos \,\alpha +i\,\sin \alpha ,\,{{z}_{2}}=\cos \beta +i\,\sin \beta \] \[\frac{({{z}_{1}}-{{z}_{2}})\,({{z}_{1}}{{z}_{2}}+1)}{({{z}_{1}}+{{z}_{2}})\,({{z}_{1}}{{z}_{2}}-1)}\] \[[(\cos \alpha -\cos \beta )+i(\sin \alpha -\sin \beta )]\] \[=\frac{[\cos \,(\alpha +\beta )+i\,\sin (\alpha +\beta )+1]}{[(\cos \alpha +\cos \beta )+i(\sin \alpha +\sin \beta )]}\] \[[\cos \,(\alpha +\beta )+i\,\sin (\alpha +\beta )-]\] \[\begin{align} & \left[ -2\sin \left( \frac{\alpha +\beta }{2} \right)\sin \left( \frac{\alpha -\beta }{2} \right)+i\,2\sin \left( \frac{\alpha -\beta }{2} \right)\cos \left( \frac{\alpha +\beta }{2} \right) \right] \\ & \\ \end{align}\] \[=\frac{\left[ 2\cos \left( \frac{\alpha +\beta }{2} \right)+2i\sin \left( \frac{\alpha +\beta }{2} \right)\cos \left( \frac{\alpha +\beta }{2} \right) \right]}{\left[ 2\cos \,\left( \frac{\alpha +\beta }{2} \right)\cos \,\left( \frac{\alpha +\beta }{2} \right)+2\,i\,\sin \,\left( \frac{\alpha -\beta }{2} \right)\,\cos \,\left( \frac{\alpha -\beta }{2} \right) \right]}\] \[\left[ -2{{\sin }^{2}}\left( \frac{\alpha +\beta }{2} \right)+2i\sin \left( \frac{\alpha +\beta }{2} \right)\cos \left( \frac{\alpha -\beta }{2} \right) \right]\] \[4\sin \left( \frac{\alpha -\beta }{2} \right)\cos \,\left( \frac{\alpha +\beta }{2} \right)\,\left[ 2\sin \,\left( \frac{\alpha +\beta }{2} \right)+i\,\cos \left( \frac{\alpha +\beta }{2} \right)\, \right]\] \[=\frac{\left[ \cos \,\,\left( \frac{\alpha +\beta }{2} \right)+i\,\sin \,\left( \frac{\alpha +\beta }{2} \right) \right]}{4\,\cos \,\left( \frac{\alpha -\beta }{2} \right)-\sin \left( \frac{\alpha +\beta }{2} \right)\left[ \cos \,\left( \frac{\alpha +\beta }{2} \right)+i\sin \left( \frac{\alpha +\beta }{2} \right) \right]}\] \[\left[ -\sin \left( \frac{\alpha +\beta }{2} \right)+i\cos \left( \frac{\alpha +\beta }{2} \right) \right]\] \[=\tan \left( \frac{\alpha -\beta }{2} \right)\,\cot \,\left( \frac{\alpha +\beta }{2} \right)\]
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