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Question Number 31549 by de best last updated on 10/Mar/18
witout using calculator, simpify  cos22.5°
$${witout}\:{using}\:{calculator},\:{simpify} \\ $$$${cos}\mathrm{22}.\mathrm{5}° \\ $$
Answered by $@ty@m last updated on 10/Mar/18
cos 2x=2cos^2 x−1  ⇒cos x=(√((1+cos 2x)/2))  ∴cos 22.5^o =(√((1+cos 45^o )/2))  =(√((1+(1/( (√2))))/2))=(√(((√2)+1)/(2(√2))))
$$\mathrm{cos}\:\mathrm{2}{x}=\mathrm{2cos}\:^{\mathrm{2}} {x}−\mathrm{1} \\ $$$$\Rightarrow\mathrm{cos}\:{x}=\sqrt{\frac{\mathrm{1}+\mathrm{cos}\:\mathrm{2}{x}}{\mathrm{2}}} \\ $$$$\therefore\mathrm{cos}\:\mathrm{22}.\mathrm{5}^{{o}} =\sqrt{\frac{\mathrm{1}+\mathrm{cos}\:\mathrm{45}^{{o}} }{\mathrm{2}}} \\ $$$$=\sqrt{\frac{\mathrm{1}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}{\mathrm{2}}}=\sqrt{\frac{\sqrt{\mathrm{2}}+\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}} \\ $$
Commented by MJS last updated on 10/Mar/18
=((√(2+(√2)))/2)
$$=\frac{\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}}{\mathrm{2}} \\ $$

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