Evaluate $\dfrac{5\cos^2 60^\circ + 4\sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$
Generated by claude-sonnet-4-6 · 2026-06-15 10:24 · grounding rag
Model Answer
Known values: $\cos 60° = \dfrac{1}{2}$, $\sec 30° = \dfrac{2}{\sqrt{3}}$, $\tan 45° = 1$, $\sin^2 30° + \cos^2 30° = 1$
$$= \frac{5\left(\dfrac{1}{2}\right)^2 + 4\left(\dfrac{2}{\sqrt{3}}\right)^2 - (1)^2}{1}$$
$$= 5 \times \frac{1}{4} + 4 \times \frac{4}{3} - 1$$
$$= \frac{5}{4} + \frac{16}{3} - 1 = \frac{15}{12} + \frac{64}{12} - \frac{12}{12} = \frac{67}{12}$$
Source: Exercise 8.2(v), Section 8.3, Chapter 8
Explanation
- The denominator simplifies to 1 using the identity $\sin^2\theta + \cos^2\theta = 1$ — a common examiner trap; recognise it immediately.
- Substitute standard values from Table 8.1 carefully: $\sec^2 30° = \left(\dfrac{2}{\sqrt{3}}\right)^2 = \dfrac{4}{3}$, not $\dfrac{2}{3}$.
- Take LCM of 4 and 3 (= 12) for the final addition. The answer $\dfrac{67}{12}$ must be left as an improper fraction.