Prove the equality :
√33−16√3sin80∘=1+8sin10∘
First, note that
cos220∘=cos20∘(2⋅12cos20∘)=cos20∘(2cos60∘cos20∘)=cos20∘((cos(60∘+20∘)+cos(60∘−20∘))=cos20∘(cos80∘+cos40∘)=cos20∘cos80∘+cos20∘cos40∘=12(cos(20∘+80∘)+cos(80∘−20∘))+12(cos(20∘+40∘)+cos(40∘−20∘))=12(cos100∘+cos60∘)+12(cos60∘+cos20∘)=12(cos(90∘+10∘)+12)+12(12+cos20∘)=12(−sin10∘)+14+14+12cos20∘=12(cos20∘−sin10∘+1)
∴
Next, we work from the LHS of the given equality and see that
\begin{align*}\sqrt{33 − 16\sqrt{3}\sin80^\circ}&=\sqrt{33-32\left(\dfrac{\sqrt{3}}{2}\right)\sin 80^\circ}\\&=\sqrt{33-32\sin 60^\circ\sin 80^\circ}\\&=\sqrt{33-16(2\sin 60^\circ\sin 80^\circ)}\\&=\sqrt{33-16\left(-\cos( 60^\circ+80^\circ)+\cos( 80^\circ-60^\circ)\right)}\\&=\sqrt{33-16\left(-\cos(140^\circ)+\cos(20^\circ)\right)}\\&=\sqrt{33-16\left(-\cos(180^\circ-40^\circ)+\cos(20^\circ)\right)}\\&=\sqrt{33-16\left(\cos(40^\circ)+\cos(20^\circ)\right)}\\&=\sqrt{33-16\cos(40^\circ)-16\cos(20^\circ)}\\&=\sqrt{33-16(2\cos^2(20^\circ)-1)-16\cos(20^\circ)}\\&=\sqrt{33-16(\cos20^\circ-\sin10^\circ+1-1)-16\cos(20^\circ)}\\&=\sqrt{33-32\cos20^\circ+16\sin10^\circ}\\&=\sqrt{33-32(1-2\sin^2 10^\circ)+16\sin10^\circ}\\&=\sqrt{1+16\sin10^\circ+64\sin^2 10^\circ)}\\&=\sqrt{(1+8\sin10^\circ)^2}\\&=1+8\sin10^\circ\end{align*}
and we're hence done.
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