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Question Number 208437 by efronzo1 last updated on 16/Jun/24

Answered by som(math1967) last updated on 16/Jun/24

$$a=sin84+cos66$$$$\Rightarrow a=sin84+sin24$$$$\Rightarrow a=2sin54cos30$$$$\therefore a=2\times \frac{\sqrt{5}+1}{4}\times \frac{\sqrt{3}}{2}=\frac{\sqrt{3}(\sqrt{5}+1)}{4}$$$$b=sin48-sin12$$$$\Rightarrow b=2sin18.cos30=\frac{\sqrt{3}(\sqrt{5}-1)}{4}$$$$ab=\frac{3(5-1)}{4\times 4}=\frac{3}{4}$$$$\frac{a}{b}=\frac{\sqrt{5}+1}{\sqrt{5}-1}$$$$\Rightarrow \frac{a+b}{a-b}=\frac{2\sqrt{5}}{2}=\sqrt{5}$$$${\left(\frac{a+b}{a-b}\right)}^{8ab}=5^{\frac{1}{2}\times 8\times \frac{3}{4}}=5^3$$

Commented by efronzo1 last updated on 16/Jun/24

$$=5^3$$

Commented by som(math1967) last updated on 16/Jun/24

$$yes$$

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