Let ( + )=-(1)/(10) and ( - )=(3)/(8), where 0

Question

Let $\cos (\alpha +\beta )=-\frac{1}{10}$ and $\sin (\alpha -\beta )=\frac{3}{8}$, where $0<\alpha <\frac{\pi }{3}$ and $0<\beta <\frac{\pi }{4}$. If $	an 2\alpha =\frac{3(1-r\sqrt{5})}{\sqrt{11}(s+\sqrt{5})},r,s\in N$, then $r+s$ is equal to $____$ .

Answer: 20

Accepted Answer

$\begin{gathered} 	an 2\alpha =	an [(\alpha +\beta )+(\alpha -\beta )] \ 	an 2\alpha =\frac{	an (\alpha +\beta )+	an (\alpha -\beta )}{1-	an (\alpha +\beta )\cdot 	an (\alpha -\beta )} \ 	an 2\alpha =\frac{(-\sqrt{99}+\frac{3}{\sqrt{55}})}{1-(\sqrt{99})(\frac{3}{\sqrt{55}})} \ 	an 2\alpha =\frac{-3\sqrt{11}+\frac{3}{\sqrt{5}	imes \sqrt{11}}}{1+\frac{9\sqrt{11}}{\sqrt{5}	imes \sqrt{11}}} \ 	an 2\alpha =\frac{3(1-11\sqrt{5})}{\sqrt{11}(9+\sqrt{5})} \ r=11,s=9 \ r+s=20 \end{gathered}$

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