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Question Number 211019 by peter frank last updated on 26/Aug/24

Answered by som(math1967) last updated on 26/Aug/24

$$letx=sin\alpha y=sin\beta$$$$cos\alpha +cos\beta =a(sin\alpha -sin\beta )$$$$\Rightarrow \frac{2cos\frac{\alpha +\beta }{2}cos\frac{\alpha -\beta }{2}}{2sin\frac{\alpha -\beta }{2}cos\frac{\alpha +\beta }{2}}=a$$$$\Rightarrow cot\frac{\alpha -\beta }{2}=a$$$$\Rightarrow \frac{1}{2}(\alpha -\beta )={\cot }^{-1}a$$$$\Rightarrow {\sin }^{-1}x-{\sin }^{-1}y={2\cot }^{-1}a$$$$\Rightarrow \frac{1}{\sqrt{1-x^2}}-\frac{1}{\sqrt{1-y^2}}\times \frac{dy}{dx}=0$$$$\therefore \frac{dy}{dx}=\sqrt{\frac{1-y^2}{1-x^2}}$$

Commented by peter frank last updated on 27/Aug/24

$$\mathrm{thank}\mathrm{you}\mathrm{som}$$

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