Q1(v):

Evaluate the following : (v) $\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$

Given: The trigonometric expression $\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$.

To Find: The numerical value of the given expression.

Step 1: Identify the values of the trigonometric ratios.
Based on the standard trigonometric table for specific angles, we have:
$\cos 60^\circ = \frac{1}{2}$
$\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}}$
$\tan 45^\circ = 1$
$\sin 30^\circ = \frac{1}{2}$
$\cos 30^\circ = \frac{\sqrt{3}}{2}$

Step 2: Substitute the values into the numerator.
The numerator is $5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ$.
Substituting the values:
$= 5 \left( \frac{1}{2} \right)^2 + 4 \left( \frac{2}{\sqrt{3}} \right)^2 - (1)^2$
$= 5 \left( \frac{1}{4} \right) + 4 \left( \frac{4}{3} \right) - 1$
$= \frac{5}{4} + \frac{16}{3} - 1$

Step 3: Simplify the numerator.
To add the fractions, find the least common multiple (LCM) of 4 and 3, which is 12.
$= \frac{5 \times 3}{4 \times 3} + \frac{16 \times 4}{3 \times 4} - \frac{1 \times 12}{1 \times 12}$
$= \frac{15}{12} + \frac{64}{12} - \frac{12}{12}$
$= \frac{15 + 64 - 12}{12}$
$= \frac{67}{12}$

Step 4: Simplify the denominator.
The denominator is $\sin^2 30^\circ + \cos^2 30^\circ$.
[Using the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$ for any angle $\theta$]:
$\sin^2 30^\circ + \cos^2 30^\circ = 1$
Alternatively, by calculation:
$= \left( \frac{1}{2} \right)^2 + \left( \frac{\sqrt{3}}{2} \right)^2$
$= \frac{1}{4} + \frac{3}{4} = \frac{4}{4} = 1$

Step 5: Calculate the final value.
The expression is $\frac{\text{Numerator}}{\text{Denominator}} = \frac{67/12}{1}$.
$= \frac{67}{12}$

Final Answer: \frac{67}{12}