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

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