Trigonometry & Inverse Trigonometry
Trig Ratios Functions Identities
nta_abhyas_2025
Grade 11
Question:
If $A$, $B$, $C$ are in arithmetic progression and $B = \frac{\pi}{4}$, then $A \tan B \tan C =$
Step-by-Step Solution
Key Concept: Arithmetic progression of angles combined with tangent product identities yields a constant value independent of $\theta$.
Given angles $A$, $B$, $C$ are in arithmetic progression with $\angle B = \frac{\pi}{3}$. Then $A = \frac{\pi}{3} - \theta$ and $C = \frac{\pi}{3} + \theta$. We evaluate $\tan\left(\frac{\pi}{3} - \theta\right) \tan\left(\frac{\pi}{3} + \theta\right) = \frac{1 - \tan \theta}{1 + \tan \theta} \cdot \frac{1 + \tan \theta}{1 - \tan \theta} = 1$.
Correct Answer: 1