Question Number 208437 by efronzo1 last updated on 16/Jun/24
Answered by som(math1967) last updated on 16/Jun/24
$$\:{a}={sin}\mathrm{84}+{cos}\mathrm{66} \\ $$$$\Rightarrow{a}={sin}\mathrm{84}+{sin}\mathrm{24} \\ $$$$\Rightarrow{a}=\mathrm{2}{sin}\mathrm{54}{cos}\mathrm{30} \\ $$$$\:\therefore{a}=\mathrm{2}×\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{4}}×\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}=\frac{\sqrt{\mathrm{3}}\left(\sqrt{\mathrm{5}}+\mathrm{1}\right)}{\mathrm{4}} \\ $$$${b}={sin}\mathrm{48}−{sin}\mathrm{12} \\ $$$$\Rightarrow{b}=\mathrm{2}{sin}\mathrm{18}.{cos}\mathrm{30}=\frac{\sqrt{\mathrm{3}}\left(\sqrt{\mathrm{5}}−\mathrm{1}\right)}{\mathrm{4}} \\ $$$${ab}=\frac{\mathrm{3}\left(\mathrm{5}−\mathrm{1}\right)}{\mathrm{4}×\mathrm{4}}=\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\:\frac{{a}}{{b}}=\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\:\sqrt{\mathrm{5}}−\mathrm{1}} \\ $$$$\Rightarrow\frac{{a}+{b}}{{a}−{b}}=\frac{\mathrm{2}\sqrt{\mathrm{5}}}{\mathrm{2}}=\sqrt{\mathrm{5}} \\ $$$$\:\left(\frac{{a}+{b}}{{a}−{b}}\right)^{\mathrm{8}{ab}} =\mathrm{5}^{\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{8}×\frac{\mathrm{3}}{\mathrm{4}}} =\mathrm{5}^{\mathrm{3}} \\ $$
Commented by efronzo1 last updated on 16/Jun/24
$$\:=\:\mathrm{5}^{\mathrm{3}} \\ $$
Commented by som(math1967) last updated on 16/Jun/24
$${yes} \\ $$