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Question Number 38743 by kunal1234523 last updated on 29/Jun/18

$$\mathrm{If}\sin \theta +sin\varphi =a\mathrm{and}\cos \theta +\cos \varphi =b,$$$$\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\tan \frac{\theta -\varphi }{2}\left(intermsofaandb\right).$$

Answered by math1967 last updated on 29/Jun/18

$$\sin \theta +\sin \varphi =a$$$$2\sin \frac{\theta +\varphi }{2}\cos \frac{\theta -\varphi }{2}=a$$$$\cos \theta +\cos \varphi =b$$$$2\cos \frac{\theta +\varphi }{2}\cos \frac{\theta -\varphi }{2}=b$$$$4\cos {}^2\frac{\theta -\varphi }{2}({sin}^2\frac{\theta +\varphi }{2}+{cos}^2\frac{\theta +\varphi }{2})=a^2+b^2$$$${cos}^2\frac{\theta -\varphi }{2}=\frac{a^2+b^2}{4}$$$${sec}^2\frac{\theta -\varphi }{2}=\frac{4}{a^2+b^2}$$$$\therefore tan\frac{\theta -\varphi }{2}=\sqrt{\frac{4-a^2-b^2}{a^2+b^2}}$$

Commented by kunal1234523 last updated on 29/Jun/18

$$thankyousir$$

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