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If-tan-pi-24-a-b-c-d-where-a-b-c-d-are-postive-numbers-Find-the-value-of-a-b-c-d-2-

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Question Number 194937 by dimentri last updated on 20/Jul/23
If tan ((π/(24)))= ((√a)−(√b))((√c)−(√d)) where a,b,c,d are postive numbers. Find the value of (a+b+c+d+2)
$$\:\:{If}\:\mathrm{tan}\:\left(\frac{\pi}{\mathrm{24}}\right)=\:\left(\sqrt{{a}}−\sqrt{{b}}\right)\left(\sqrt{{c}}−\sqrt{{d}}\right) \\ $$$$\:\:{where}\:{a},{b},{c},{d}\:{are}\:{postive}\:{numbers}. \\ $$$$\:\:{Find}\:{the}\:{value}\:{of}\:\left({a}+{b}+{c}+{d}+\mathrm{2}\right) \\ $$
Answered by BaliramKumar last updated on 20/Jul/23
tan((π/(24))) = tan7.5° tan15° = 2 − (√3) ((2tan7.5°)/(1−tan^2 7.5°)) = 2 − (√3) ⇒((2x)/(1−x^2 )) = 2 − (√3) x = tan7.5 = 2(√(2+(√3))) − 2 − (√3) tan7.5 = (√2)(√(4+2(√3))) − 2 − (√3) tan7.5 = (√2)((√3) + 1) − 2 − (√3) tan7.5 = (√6) + (√2) − 2 − (√3) tan7.5 = (√6) − (√3) − 2 + (√2) tan7.5 = (√3)((√2) − 1) − (√2)((√2) − 1) ((√a) − (√b))((√c) − (√d)) = ((√3) − (√2))((√2) − (√1)) a = 3, b = 2, c = 2, d = 1 a + b + c + d + 2 = determinant (((10)))
$$\mathrm{tan}\left(\frac{\pi}{\mathrm{24}}\right)\:=\:\mathrm{tan7}.\mathrm{5}° \\ $$$$\mathrm{tan15}°\:=\:\mathrm{2}\:−\:\sqrt{\mathrm{3}} \\ $$$$\frac{\mathrm{2tan7}.\mathrm{5}°}{\mathrm{1}−\mathrm{tan}^{\mathrm{2}} \mathrm{7}.\mathrm{5}°}\:=\:\mathrm{2}\:−\:\sqrt{\mathrm{3}}\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\frac{\mathrm{2x}}{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\:=\:\mathrm{2}\:−\:\sqrt{\mathrm{3}}\: \\ $$$$\mathrm{x}\:=\:\mathrm{tan7}.\mathrm{5}\:=\:\mathrm{2}\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}\:−\:\mathrm{2}\:−\:\sqrt{\mathrm{3}} \\ $$$$\:\mathrm{tan7}.\mathrm{5}\:=\:\sqrt{\mathrm{2}}\sqrt{\mathrm{4}+\mathrm{2}\sqrt{\mathrm{3}}}\:−\:\mathrm{2}\:−\:\sqrt{\mathrm{3}} \\ $$$$\:\mathrm{tan7}.\mathrm{5}\:=\:\sqrt{\mathrm{2}}\left(\sqrt{\mathrm{3}}\:+\:\mathrm{1}\right)\:−\:\mathrm{2}\:−\:\sqrt{\mathrm{3}} \\ $$$$\:\mathrm{tan7}.\mathrm{5}\:=\:\sqrt{\mathrm{6}}\:+\:\sqrt{\mathrm{2}}\:−\:\mathrm{2}\:−\:\sqrt{\mathrm{3}} \\ $$$$\:\mathrm{tan7}.\mathrm{5}\:=\:\sqrt{\mathrm{6}}\:−\:\:\sqrt{\mathrm{3}}\:−\:\mathrm{2}\:+\:\sqrt{\mathrm{2}} \\ $$$$\:\mathrm{tan7}.\mathrm{5}\:=\:\sqrt{\mathrm{3}}\left(\sqrt{\mathrm{2}}\:−\:\mathrm{1}\right)\:−\:\sqrt{\mathrm{2}}\left(\sqrt{\mathrm{2}}\:−\:\mathrm{1}\right) \\ $$$$\left(\sqrt{\mathrm{a}}\:−\:\sqrt{\mathrm{b}}\right)\left(\sqrt{\mathrm{c}}\:−\:\sqrt{\mathrm{d}}\right)\:=\:\left(\sqrt{\mathrm{3}}\:−\:\sqrt{\mathrm{2}}\right)\left(\sqrt{\mathrm{2}}\:−\:\sqrt{\mathrm{1}}\right) \\ $$$$\mathrm{a}\:=\:\mathrm{3},\:\:\:\:\:\mathrm{b}\:=\:\mathrm{2},\:\:\:\:\:\:\mathrm{c}\:=\:\mathrm{2},\:\:\:\:\:\:\:\:\mathrm{d}\:=\:\mathrm{1} \\ $$$$\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:+\:\mathrm{d}\:+\:\mathrm{2}\:=\:\begin{array}{|c|}{\mathrm{10}}\\\hline\end{array} \\ $$$$ \\ $$

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