If f(x)
Question:

If $f(x)=\frac{\sin ^{4} x+\cos ^{2} x}{\sin ^{2} x+\cos ^{4} x}$ for $x \in \mathrm{R}$, then $f(2002)=$

(a) 1

(b) 2

(c) 3

(d) 4

Solution:

(a) 1

Given:

$f(x)=\frac{\sin ^{4} x+\cos ^{2} x}{\sin ^{2} x+\cos ^{4} x}$

On dividing the numerator and denominator by $\cos ^{4}$

$f(x)=\frac{\tan ^{4} x+\sec ^{2} x}{1+\tan ^{2} x \sec ^{2} x}=\frac{1+\tan ^{4} x+\tan ^{2} x}{1+\tan ^{2} x\left(1+\tan ^{2} x\right)}=\frac{1+\tan ^{4} x+\tan ^{2} x}{1+\tan ^{4} x+\tan ^{2} x}=1$ (For every $x \in \mathrm{R}$ )

For = 2002, we have

f (2002) = 1