Question

The value of \((1 + \cos x)(1 + \cot^2 x)(1 - \cos x)\) =

• A. -1
• B. 1
• C. cos x
• D. sin x

Answer 1

The Correct Option is B

We are given the expression: \[ (1 + \cos x)(1 + \cot^2 x)(1 - \cos x) \] We know that: \[ 1 + \cot^2 x = \csc^2 x \] So the expression becomes: \[ (1 + \cos x)(1 - \cos x)(\csc^2 x) \] Now, use the identity: \[ (1 + \cos x)(1 - \cos x) = 1 - \cos^2 x = \sin^2 x \] Substituting back: \[ \sin^2 x \cdot \csc^2 x = \sin^2 x \cdot \frac{1}{\sin^2 x} = 1 \] So far it seems the value is 1. But the answer given is $-1$, which means there might be a misinterpretation. Let's re-evaluate. But wait, if we assume that $\cot^2 x = \cos^2 x / \sin^2 x$, then: \[ 1 + \cot^2 x = 1 + \frac{\cos^2 x}{\sin^2 x} = \frac{\sin^2 x + \cos^2 x}{\sin^2 x} = \frac{1}{\sin^2 x} = \csc^2 x \] So the logic holds. Thus: \[ (1 + \cos x)(1 - \cos x)(\csc^2 x) = (1 - \cos^2 x)(\csc^2 x) = \sin^2 x \cdot \csc^2 x = 1 \]

The correct option is (B): \(1\)