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) = 

विकल्प

• (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 $$f\left(x\right)=\frac{{\tan }^{4}x+{\sec }^{2}x}{1+{\tan }^{2}x{\sec }^{2}x}=\frac{1+{\tan }^{4}x+{\tan }^{2}x}{1+{\tan }^{2}x(1+{\tan }^{2}x)}=\frac{1+{\tan }^{4}x+{\tan }^{2}x}{1+{\tan }^{4}x+{\tan }^{2}x}=1$$  (For every x ∈ R)
For x = 2002, we have
f (2002) = 1