If $4k = \tan^2 60^\circ - 2\cosec^2 30^\circ - 2\tan^2 30^\circ$, then find the value of $k$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:22 · grounding rag
Model Answer
Given: $4k = \tan^2 60° - 2\cosec^2 30° - 2\tan^2 30°$
Substituting values: $\tan 60° = \sqrt{3}$, $\cosec 30° = 2$, $\tan 30° = \dfrac{1}{\sqrt{3}}$
$$4k = (\sqrt{3})^2 - 2(2)^2 - 2\left(\frac{1}{\sqrt{3}}\right)^2$$
$$4k = 3 - 2(4) - 2\left(\frac{1}{3}\right)$$
$$4k = 3 - 8 - \frac{2}{3} = -5 - \frac{2}{3} = \frac{-17}{3}$$
$$k = \frac{-17}{12}$$
Source: Chapter 8, Section 8.3 (Table 8.1)
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Explanation
- Examiners expect you to first write down the standard values clearly, then substitute and simplify step by step — each step can carry partial marks.
- Key values to memorise: $\tan 60° = \sqrt{3}$, $\cosec 30° = 2$, $\tan 30° = \frac{1}{\sqrt{3}}$.
- Don't skip the squaring step; write $(\sqrt{3})^2 = 3$, $(2)^2 = 4$, $\left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}$ explicitly to avoid sign/arithmetic errors.