If f(x)= ^ 4 x+ ^2x ^2x+ ^4x for x , then f(2002) = - Vidyadip

MCQ

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

✓
$1$
B
$2$
C
$3$
D
$4$

✓

Answer

Correct option: A.

$1$

Given,
$\text{f(x)}=\frac{\sin^{4}\text{x}+\cos^2\text{x}}{\sin^2\text{x}+\cos^4\text{x}}$
On dividing the numerator and denominator by $\cos^4\text{x},$ we get
$\text{f(x)}=\frac{\tan^4\text{x}+\sec^2\text{x}}{1+\tan^2\text{x}\sec^2\text{x}}$
$=\frac{1+\tan^4\text{x}+\tan^2\text{x}}{1+\tan^2\text{x}(1+\tan^2\text{x})}$
$=\frac{1+\tan^{4}\text{x}+\tan^{2}\text{x}}{1+\tan^{4}\text{x}+\tan^{2}\text{x}}=1$
$($For every $x\in\text{R})$
$\text{For x}=2002,$
We have,
$\text{f}(2002)=1$

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