If , then  is

Given:

sec() = a

Concept Used:

Implicit differentiation.

Chain rule of differentiation.

Calculation:

⇒ = a

⇒ cos() =

Differentiate both sides with respect to x:

⇒ -sin() × () = 0

⇒ -sin() × = 0

Since sin() is unlikely to be zero (as that would make 'a' undefined), we can focus on the other term:

⇒ (x+y)(1-) - (x-y)(1+) = 0

⇒ x + y - x - y - x + y - x + y = 0

⇒ 2y - 2x = 0

⇒ 2x = 2y

⇒ =

∴ =

Hence option 4 is correct