Introduction to Trigonometry
RD Sharma
CBSE
Grade 10
Question:
If $\sin \theta + \cos \theta = \sqrt{3}$, prove that $\tan \theta + \cot \theta = 1$.
Step-by-Step Solution
Key Concept: Square both sides: $(\sin \theta + \cos \theta)^2 = 3 \Rightarrow 1 + 2 \sin \theta \cos \theta = 3 \Rightarrow 2 \sin \theta \cos \theta = 2 \Rightarrow \sin \theta \cos \theta = 1$. $\tan \theta + \cot \theta = \dfrac{\sin \theta}{\cos \theta} + \dfrac{\cos \theta}{\sin \theta} = \dfrac{1}{\sin \theta \cos \theta} = \dfrac{1}{1} = 1$.
$(\sin\theta + \cos\theta)^2 = 3 \Rightarrow 1 + 2\sin\theta\cos\theta = 3 \Rightarrow \sin\theta\cos\theta = 1$. [1.5 Marks]
$\tan\theta + \cot\theta = \dfrac{\sin^2\theta + \cos^2\theta}{\sin\theta\cos\theta} = \dfrac{1}{\sin\theta\cos\theta} = \dfrac{1}{1} = 1$. Proved! [1.5 Marks]
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🎯 Official CBSE Marking Scheme:
Finding $\sin\theta\cos\theta = 1$: 1.5 Marks
Evaluating $\tan\theta + \cot\theta = 1$: 1.5 Marks
Correct Answer:
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