Q1. [1] § 8.3 Trigonometric Ratios of Some Specific Angles § 8.5 Summary
$\frac{1 - \tan^2 30°}{1 + \tan^2 30°}$ is equal to
- A $\sin 60°$
- B $\cos 60°$
- C $\tan 60°$
- D $\sec 60°$
Previously asked in CBSE board exam
2025 30/5/1 Q12
Generated by claude-sonnet-4-6 · 2026-06-15 10:26 · grounding rag
Model Answer
(B) cos 60°
$\tan 30° = \dfrac{1}{\sqrt{3}}$, so $\tan^2 30° = \dfrac{1}{3}$.
$$\frac{1 - \frac{1}{3}}{1 + \frac{1}{3}} = \frac{\frac{2}{3}}{\frac{4}{3}} = \frac{1}{2} = \cos 60°$$
Source: Chapter 8, Section 8.3 (Table 8.1)
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Explanation
- Substitute $\tan 30° = \frac{1}{\sqrt{3}}$ from the standard table, then simplify the fraction.
- The result $\frac{1}{2}$ matches $\cos 60°$, not $\sin 60° = \frac{\sqrt{3}}{2}$ or $\tan 60° = \sqrt{3}$.
- This is the double-angle cosine formula: $\cos 2\theta = \frac{1-\tan^2\theta}{1+\tan^2\theta}$; here $\theta = 30°$, giving $\cos 60°$.
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