Our question that we have at hand here is 8(cos 80°)(cos 140°)(cos 160°). We need to convert that into 'decimal form' to prove that it equals 1. However decimal form doesn't generally give us a decimal, in this case.
Let's start by using the identity cos(s)cos(t) = cos(s + t) + cos(s - t) / 2. In this case we are looking particularly at the expression (cos 80°)(cos 140°), where s = 80, and t = 140.
[tex]\cos \left(80^{\circ \:}\right)\cos \left(140^{\circ \:}\right)=\frac{\cos \left(80^{\circ \:}+140^{\circ \:}\right)+\cos \left(80^{\circ \:}-140^{\circ \:}\right)}{2}[/tex]
[tex]=8\cdot \frac{\cos \left(80^{\circ \:}+140^{\circ \:}\right)+\cos \left(80^{\circ \:}-140^{\circ \:}\right)}{2}\cos \left(160^{\circ \:}\right)[/tex]
[tex]= 4\cos \left(160^{\circ \:}\right)\left(\cos \left(220^{\circ \:}\right)+\cos \left(-60^{\circ \:}\right)\right)[/tex]
To further simplify this expression, we know that cos(- 60°) = cos(60°). This may not seem like a very effective step, but we can apply a few trivial identities here. Remember that cos(60°) = 1 / 2,
[tex]4\cos \left(160^{\circ \:}\right)\left(\cos \left(220^{\circ \:}\right)+\frac{1}{2}\right)[/tex]
[tex]= 4(0.25)\\= 1[/tex]
And hence we proved that 8(cos 80°)(cos 140°)(cos 160°) in decimal form, = 1.
