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Question Number 194937 by dimentri last updated on 20/Jul/23

$$If\tan \left(\frac{\pi }{24}\right)=(\sqrt{a}-\sqrt{b})(\sqrt{c}-\sqrt{d})$$$$wherea,b,c,darepostivenumbers.$$$$Findthevalueof(a+b+c+d+2)$$

Answered by BaliramKumar last updated on 20/Jul/23

$$\tan \left(\frac{\pi }{24}\right)=\tan 7.5°$$$$\tan 15°=2-\sqrt{3}$$$$\frac{2\tan 7.5°}{1-{\tan }^{2}7.5°}=2-\sqrt{3}\Rightarrow \frac{2\mathrm{x}}{1-{\mathrm{x}}^2}=2-\sqrt{3}$$$$\mathrm{x}=\tan 7.5=2\sqrt{2+\sqrt{3}}-2-\sqrt{3}$$$$\tan 7.5=\sqrt{2}\sqrt{4+2\sqrt{3}}-2-\sqrt{3}$$$$\tan 7.5=\sqrt{2}(\sqrt{3}+1)-2-\sqrt{3}$$$$\tan 7.5=\sqrt{6}+\sqrt{2}-2-\sqrt{3}$$$$\tan 7.5=\sqrt{6}-\sqrt{3}-2+\sqrt{2}$$$$\tan 7.5=\sqrt{3}(\sqrt{2}-1)-\sqrt{2}(\sqrt{2}-1)$$$$(\sqrt{\mathrm{a}}-\sqrt{\mathrm{b}})(\sqrt{\mathrm{c}}-\sqrt{\mathrm{d}})=(\sqrt{3}-\sqrt{2})(\sqrt{2}-\sqrt{1})$$$$\mathrm{a}=3,\mathrm{b}=2,\mathrm{c}=2,\mathrm{d}=1$$$$\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+2=\overline{)\begin{array}{c}10\end{array}}$$$$$$

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