Question Number 108114 by mathdave last updated on 14/Aug/20
Answered by mr W last updated on 14/Aug/20
$$\mathrm{sin}\:\left(\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{4}}{\mathrm{5}}−\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{4}+\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{10}}\right) \\ $$$$=\frac{\mathrm{4}}{\mathrm{5}}×\frac{\mathrm{4}+\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{10}}−\frac{\mathrm{3}}{\mathrm{5}}×\frac{\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{3}}{\mathrm{10}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{4}}{\mathrm{5}}−\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{4}+\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{10}}=\frac{\pi}{\mathrm{6}} \\ $$