Question Number 31549 by de best last updated on 10/Mar/18
$${witout}\:{using}\:{calculator},\:{simpify} \\ $$$${cos}\mathrm{22}.\mathrm{5}° \\ $$
Answered by $@ty@m last updated on 10/Mar/18
$$\mathrm{cos}\:\mathrm{2}{x}=\mathrm{2cos}\:^{\mathrm{2}} {x}−\mathrm{1} \\ $$$$\Rightarrow\mathrm{cos}\:{x}=\sqrt{\frac{\mathrm{1}+\mathrm{cos}\:\mathrm{2}{x}}{\mathrm{2}}} \\ $$$$\therefore\mathrm{cos}\:\mathrm{22}.\mathrm{5}^{{o}} =\sqrt{\frac{\mathrm{1}+\mathrm{cos}\:\mathrm{45}^{{o}} }{\mathrm{2}}} \\ $$$$=\sqrt{\frac{\mathrm{1}+\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}}{\mathrm{2}}}=\sqrt{\frac{\sqrt{\mathrm{2}}+\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}} \\ $$
Commented by MJS last updated on 10/Mar/18
$$=\frac{\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}}{\mathrm{2}} \\ $$