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Question Number 13249 by Tinkutara last updated on 17/May/17

$$\mathrm{If}m\sin \theta =n\sin (\theta +2\alpha ),\mathrm{then}$$$$\tan (\theta +\alpha )\cot \alpha \mathrm{equal}\mathrm{to}$$$$\left[\mathrm{Answer}\mathrm{given}\mathrm{in}\mathrm{my}\mathrm{book}\mathrm{is}\frac{1-n}{1+n}\right]$$

Commented by prakash jain last updated on 17/May/17

$$\tan (\theta +\alpha )\cot \alpha$$$$=\frac{\sin (\theta +\alpha )}{\cos (\theta +\alpha )}.\frac{\cos \alpha }{\sin \alpha }$$$$\sin A\cos B=\frac{1}{2}(\sin (A+B)+\sin (A-B))$$$$=\frac{\frac{\sin (\theta +2\alpha )+\sin \theta }{2}}{\frac{\sin (\theta +2\alpha )+\sin (-\theta )}{2}}$$$$=\frac{\sin (\theta +2\alpha )+\sin \theta }{\sin (\theta +2\alpha )-\sin \theta }$$$$assumingsin\theta \ne 0$$$$=\frac{\frac{\sin (\theta +2\alpha )}{\sin \theta }+1}{\frac{\sin (\theta +2\alpha )}{\sin \theta }-1}=\frac{m+n}{m-n}$$$$\mathrm{The}\mathrm{answer}\mathrm{does}\mathrm{not}\mathrm{match}\mathrm{with}$$$$\mathrm{ur}\mathrm{book}.\mathrm{I}\mathrm{will}\mathrm{recheck}\mathrm{later}.$$

Commented by Tinkutara last updated on 18/May/17

$$\mathrm{I}\mathrm{was}\mathrm{reaching}\mathrm{on}\mathrm{this}\mathrm{same}\mathrm{solution}.$$$$\mathrm{Maybe}\mathrm{answer}\mathrm{in}\mathrm{my}\mathrm{book}\mathrm{is}\mathrm{wrong}.$$

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