If $4\cot^2 45° - \sec^2 60° + \sin^2 60° + p = \frac{3}{4}$, then find the value of $p$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:23 · grounding rag
Model Answer
Known values: $\cot 45° = 1$, $\sec 60° = 2$, $\sin 60° = \dfrac{\sqrt{3}}{2}$
Substituting in the equation:
$$4(1)^2 - (2)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 + p = \frac{3}{4}$$
$$4 - 4 + \frac{3}{4} + p = \frac{3}{4}$$
$$\frac{3}{4} + p = \frac{3}{4}$$
$$\boxed{p = 0}$$
Source: Introduction to Trigonometry, Section 8.3
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Explanation
- Substitute standard values directly from Table 8.1: $\cot 45°=1$, $\sec 60°=2$, $\sin 60°=\frac{\sqrt{3}}{2}$.
- The terms $4(1)-4$ cancel to zero, leaving $\frac{3}{4}+p=\frac{3}{4}$, so $p=0$.
- Examiners award 1 mark for correct substitution and 1 mark for the final value of $p$. Show each step clearly.