NIMCET 2011 — Mathematics PYQ

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Question

NIMCET 2011 — Mathematics PYQ

NIMCET | Mathematics | 2011

If $\sin x, \cos x$ and $\tan x$ are in GP, then $\cot^6 x - \cot^2 x$ will be equal to:

Choose the correct answer:

Explanation

1. Apply GP Condition:

Since the terms are in Geometric Progression:

\cos^2 x = \sin x \cdot \tan x

\cos^2 x = \sin x \cdot \frac{\sin x}{\cos x}

\cos^3 x = \sin^2 x

2. Express in terms of $\cot x$ :

Divide both sides by $\sin^3 x$ :

\frac{\cos^3 x}{\sin^3 x} = \frac{\sin^2 x}{\sin^3 x}

\cot^3 x = \frac{1}{\sin x}

3. Square both sides:

\cot^6 x = \frac{1}{\sin^2 x}

Using the identity $\frac{1}{\sin^2 x} = \csc^2 x$ :

\cot^6 x = 1 + \cot^2 x

4. Find the required value:

\cot^6 x - \cot^2 x = 1

Correct Option: (c)