Given:
Expression to evaluate:

\(\frac{7}{2}\cos{70^\circ} \div \sin{20^\circ} - \frac{4}{7}\cos{53^\circ}\csc{37^\circ} \div \tan{15^\circ}\tan{35^\circ}\tan{55^\circ}\tan{75^\circ}\)

Solution:

Step 1: Simplify the expression
To simplify the given expression, we will start by applying trigonometric identities and simplifying each term.

Term 1: \(\frac{7}{2}\cos{70^\circ} \div \sin{20^\circ}\)

We know that \(\csc{\theta} = \frac{1}{\sin{\theta}}\). Therefore, we can rewrite the expression as:

\(\frac{7}{2}\cos{70^\circ} \div \sin{20^\circ} = \frac{7}{2}\cos{70^\circ} \times \csc{20^\circ}\)

Using the identity \(\sin{(90^\circ - \theta)} = \cos{\theta}\), we can simplify further:

\(\frac{7}{2}\cos{70^\circ} \times \csc{20^\circ} = \frac{7}{2}\cos{70^\circ} \times \csc{(90^\circ - 20^\circ)}\)

\(\frac{7}{2}\cos{70^\circ} \times \csc{(90^\circ - 20^\circ)} = \frac{7}{2}\cos{70^\circ} \times \csc{70^\circ}\)

Term 2: \(- \frac{4}{7}\cos{53^\circ}\csc{37^\circ}\)

Similar to the previous term, we can rewrite the expression using the identity \(\csc{\theta} = \frac{1}{\sin{\theta}}\):

\(- \frac{4}{7}\cos{53^\circ}\csc{37^\circ} = - \frac{4}{7}\cos{53^\circ} \times \frac{1}{\sin{37^\circ}}\)

Term 3: \(\tan{15^\circ}\tan{35^\circ}\tan{55^\circ}\tan{75^\circ}\)

Using the identity \(\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}\), we can rewrite the expression as:

\(\tan{15^\circ}\tan{35^\circ}\tan{55^\circ}\tan{75^\circ} = \frac{\sin{15^\circ}}{\cos{15^\circ}} \times \frac{\sin{35^\circ}}{\cos{35^\circ}} \times \frac{\sin{55^\circ}}{\cos{55^\circ}} \times \frac{\sin{75^\circ}}{\cos{75^\circ}}\)

Step 2: Substitute values for trigonometric functions
Now, we can substitute the values of the trigonometric functions using the values of the angles provided.

\(\frac{7}{2}\cos