Trigonometric Expression Solution

We need to evaluate the given trigonometric expression:

$ E = \frac{\sqrt{3}\operatorname{cosec}20^\circ - \sec 20^\circ}{\cos 20^\circ \cos 40^\circ \cos 60^\circ \cos 80^\circ} $

Numerator Calculation

Let the numerator be $N = \sqrt{3}\operatorname{cosec}20^\circ - \sec 20^\circ$.

Expressing in terms of $\sin$ and $\cos$:

$ N = \frac{\sqrt{3}}{\sin 20^\circ} - \frac{1}{\cos 20^\circ} $

Find a common denominator:

$ N = \frac{\sqrt{3}\cos 20^\circ - \sin 20^\circ}{\sin 20^\circ \cos 20^\circ} $

Multiply the numerator and denominator by 2 to use trigonometric identities:

$ N = \frac{2(\frac{\sqrt{3}}{2}\cos 20^\circ - \frac{1}{2}\sin 20^\circ)}{2\sin 20^\circ \cos 20^\circ} $

Apply the sine subtraction formula $\sin(A - B) = \sin A \cos B - \cos A \sin B$ to the numerator, with $A=60^\circ$ and $B=20^\circ$ ($\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\cos 60^\circ = \frac{1}{2}$):

$ N = \frac{2\sin(60^\circ - 20^\circ)}{\sin(2 \times 20^\circ)} $

Simplify using the double angle formula for sine ($\sin(2A) = 2\sin A \cos A$):

$ N = \frac{2\sin 40^\circ}{\sin 40^\circ} $

$ N = 4 $

Denominator Calculation

Let the denominator be $D = \cos 20^\circ \cos 40^\circ \cos 60^\circ \cos 80^\circ$.

We know the value $\cos 60^\circ = \frac{1}{2}$.

$ D = \frac{1}{2} (\cos 20^\circ \cos 40^\circ \cos 80^\circ) $

Use the product identity $\cos x \cos(60^\circ - x) \cos(60^\circ + x) = \frac{1}{4}\cos(3x)$.

Here, $x = 20^\circ$, so $60^\circ - x = 40^\circ$ and $60^\circ + x = 80^\circ$.

$ \cos 20^\circ \cos 40^\circ \cos 80^\circ = \frac{1}{4}\cos(3 \times 20^\circ) = \frac{1}{4}\cos 60^\circ $

Substitute the value of $\cos 60^\circ = \frac{1}{2}$:

$ \cos 20^\circ \cos 40^\circ \cos 80^\circ = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} $

Now, substitute this back into the expression for $D$:

$ D = \frac{1}{2} \times \frac{1}{8} = \frac{1}{16} $

Final Value Calculation

The value of the expression $E$ is the numerator divided by the denominator:

$ E = \frac{N}{D} = \frac{4}{\frac{1}{16}} $

$ E = 4 \times 16 = 64 $