Inverse Trigonometric Functions Question - CBSE

Question

Simplify $\cot^{-1}\left(\dfrac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}\right)$, $0 < x < \dfrac{\pi}{2}$.

Accepted Answer

$\sqrt{1+\sin x} = \cos(x/2) + \sin(x/2)$ and $\sqrt{1-\sin x} = \cos(x/2) - \sin(x/2)$. [2.0 Marks]
Numerator $= 2\cos(x/2)$, Denominator $= 2\sin(x/2)$. [1.5 Marks]
Argument $= \cot(x/2) \Rightarrow \cot^{-1}(\cot(x/2)) = x/2$. [1.5 Marks]

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🎯 Official CBSE Marking Scheme:
Expressing radicals in terms of half-angle sine/cosine: 2.0 Marks
Simplifying numerator and denominator: 1.5 Marks
Evaluating simplest form $= x/2$: 1.5 Marks

Question:

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