Those Numerators Look Familiar

To get a common denominator, we have $\large{\frac{\sin 80^{o}-\sqrt{3}\cos80^{o}}{\sin80^{o}\cos80^{o}}}$ .

Then, we write the numerators in terms of trig values: $\large{2 \cdot \frac{\frac{1}{2}\sin80^{o}-\frac{\sqrt{3}}{2}\cos80^{o}}{\sin80^{o}\cos80^{o}}} = \large{2 \cdot \frac{\cos 60^{o}\sin 80^{o}-\sin 60^{o}\cos 80^{o}}{\sin80^{o}\cos80^{o}}}$

Using Sine - Sum and Difference Formulas , this is

$\large{2 \frac{\sin(80^{o}-60^{o})}{\sin80^{o}\cos80^{o}}}$

Using Double Angle Identities , this is

$\large{4 \frac{\sin20^{o}}{\sin 160^{o}}}$

Using the Fundamental Trigonometric Identity $\large{\sin(180^{o}-\theta)=\sin(\theta)}$ , this is

$\large{4 \frac{\sin20^{o}}{\sin20^{o}}}$

$\huge{=4}$

Consider the following angle formulas $\sin{3x} = 3\sin x - 4\sin ^ 3 x \\ \cos{3x} = 4\cos ^ 3 x - 3\cos x$ If we let $x = 80^o$ then $\sin 80^o ( 3 - 4\sin^2 80^o ) = \sin 240^o = -\frac{\sqrt{3}}{2} \\ \cos 80^o ( 4\cos^2 80^o - 3 ) = \cos 240^o = -\frac{1}{2}$

Now in reference to the original question. $\frac{1}{\cos 80^o} - \frac{\sqrt{3}}{\sin 80^o} = \frac{ 4\cos^2 80^o - 3 }{\cos 80^o ( 4\cos^2 80^o - 3 )} - \frac{\sqrt{3}( 3 - 4\sin^2 80^o )}{\sin 80^o ( 3 - 4\sin^2 80^o )} \\ = \frac{ 4\cos^2 80^o - 3 }{-\frac{1}{2}} + \frac{\sqrt{3}( 4\sin^2 80^o - 3 )}{-\frac{\sqrt{3}}{2}} \\ = -2(4\cos^2 80^o - 3 +4\sin^2 80^o - 3) \\ = -2(4 - 6) \\ = \boxed{4}$

$\begin{aligned} \frac{1}{\cos 80^\circ} - \frac{\sqrt{3}}{\sin 80^\circ} & = \frac{\sin 80^\circ - \sqrt{3}\cos 80^\circ}{\cos 80^\circ \sin 80^\circ} \\ & = \frac{2\left(\frac{1}{2}\sin 80^\circ - \frac{\sqrt{3}}{2}\cos 80^\circ \right)}{\frac{1}{2}\left(2\cos 80^\circ \sin 80^\circ \right)} \\ & = \frac{4(\cos 60^\circ \sin 80^\circ - \sin 60^\circ \cos 80^\circ)}{\sin 160^\circ} \\ & = \frac{4\sin (80^\circ-60^\circ)}{\sin (180^\circ-160^\circ)} \\ & = \frac{4\sin 20^\circ}{\sin 20^\circ} \\ & = \boxed{4} \end{aligned}$

1 = 2 sin 30

square root of 3 = 2 cos 30

cos 80 = sin 10

sin 80 = cos 10

Then the given expression = (2 sin 30 cos 10 - 2 sin 10 cos 30)/sin 10 cos 10 ..... (1)

2 sin 30 cos 10 - 2 sin 10 cos 30 = 2 sin (30 - 10) = 2 sin 20 .......................................... (2)

sin 10 cos 10 = (1/2) sin 20 ........................................................................................................ (3)

From (1) , (2) and (3) we get

Then the given expression = 4