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If F ( X ) = Sin 4 X + Cos 2 X Sin 2 X + Cos 4 X for X ∈ R, Then F (2002) = (A) 1 (B) 2 (C) 3 (D) 4 - CBSE (Science) Class 11 - Mathematics

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Question

If  \[f\left( x \right) = \frac{\sin^4 x + \cos^2 x}{\sin^2 x + \cos^4 x}\] for x ∈ R, then f (2002) = 

  • (a) 1

  • (b) 2

  • (c) 3

  • (d) 4

     

Solution

(a) 1
Given:

\[f\left( x \right) = \frac{\sin^4 x + \cos^2 x}{\sin^2 x + \cos^4 x}\] On dividing the numerator and denominator by \[\cos^4 x\]\ , we get \[f\left( x \right) = \frac{\tan^4 x + \sec^2 x}{1 + \tan^2 x \sec^2 x} = \frac{1 + \tan^4 x + \tan^2 x}{1 + \tan^2 x\left( 1 + \tan^2 x \right)} = \frac{1 + \tan^4 x + \tan^2 x}{1 + \tan^4 x + \tan^2 x} = 1\]  (For every x ∈ R)
For = 2002, we have
f (2002) = 1

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Solution If F ( X ) = Sin 4 X + Cos 2 X Sin 2 X + Cos 4 X for X ∈ R, Then F (2002) = (A) 1 (B) 2 (C) 3 (D) 4 Concept: Functions.
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