The number of points inside or on the circle x^2 + y^2 = 4 satisfying tan^4x + cot^4x + 1 = 3sin^2y is

Q: The number of points inside or on the circle x2 + y2 = 4 satisfying
tan4x + cot4x + 1 = 3sin2y is

(A) one

(B) two

(C) four

(D) infinite

Sol. tan4x + cot4x + 1 = (tan2x – cot2x)2 + 3 ≤ 3

3 sin2y  ≤ 3

⇒ tan2 x = cot2 x , sin2y = 1

⇒ tanx = ± 1, siny = ±1

⇒ x = ± π/4, ± 3π/4 , . . .

But x2 ≤  4 ⇒ -2 ≤  x ≤  2 ⇒ x = ± π/4 only

Siny = ± 1 ⇒  y = ± π/2 ,  3π/2 , . . ..

But y2 ≤ 4  ⇒ y = ± π/2  only.

So four solutions are possible.

Hence (C) is the correct answer.