Q112 / 200 · TS EAMCET 2002 · Mathematics

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

  • Correct answer
    A
    $1$
  • B
    $2$
  • C
    $3$
  • D
    $4$

Solution

We have, $$ f(x)=\frac{\cos ^2 x+\sin ^4 x}{\sin ^2 x+\cos ^4 x} $$ $\begin{aligned} & =\frac{1-\sin ^2 x+\sin ^4 x}{1-\cos ^2 x+\cos ^4 x} \\ & =\frac{1-\sin ^2 x\left(1-\sin ^2 x\right)}{1-\cos ^2 x\left(1-\cos ^2 x\right)} \\ & =\frac{1-\sin ^2 x \cos ^2 x}{1-\cos ^2 x \sin ^2 x}=1 \\ \therefore \quad f(2002) & =1\end{aligned}$

Hence, the correct answer is option: 'A'.

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