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) = 

Options

• (a) 1

• (b) 2

• (c) 3

• (d) 4

   

Answer 1

(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 x = 2002, we have
f (2002) = 1