If $\sqrt{3} \tan \theta = 3 \sin \theta$, find the value of $\sin^2 \theta - \cos^2 \theta$.
Step-by-Step Solution
Key Concept: Express $\tan \theta = \sin \theta / \cos \theta$, find $\cos \theta = 1/\sqrt{3}$, then compute $\sin^2 \theta - \cos^2 \theta$.
$\sqrt{3} \dfrac{\sin \theta}{\cos \theta} = 3 \sin \theta \Rightarrow \cos \theta = \dfrac{\sqrt{3}}{3} = \dfrac{1}{\sqrt{3}}$. [1.0 Mark]
$\cos^2 \theta = \dfrac{1}{3} \Rightarrow \sin^2 \theta = 1 - \dfrac{1}{3} = \dfrac{2}{3}$.
$\sin^2 \theta - \cos^2 \theta = \dfrac{2}{3} - \dfrac{1}{3} = \dfrac{1}{3}$. [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Finding $\cos \theta = 1/\sqrt{3}$: 1.0 Mark
Evaluating $\sin^2 \theta - \cos^2 \theta = 1/3$: 1.0 Mark
Correct Answer: