​ If 0 < θ, ϕ < π/2, x = sum{n=0 to ∞ } cos2n θ, y =sum{n=0 to ∞ } sin2n ϕ and z =sum{n=0 to ∞ } cos2n θ . sin2n ϕ then: ​

Question

If 0 < θ, ϕ < π/2, x = \(\sum\limits_{n=0}^\infty\) cos²ⁿ θ, y = \(\sum\limits_{n=0}^\infty\) sin²ⁿ ϕ and z = \(\sum\limits_{n=0}^\infty\) cos²ⁿ θ . sin²ⁿ ϕ then:

(1) xy - z = (x + y) z

(2) xy + yz + zx = z

(3) xyz = 4

(4) xy + z = (x + y)z

Answer 1

Correct option is (4) xy + z = (x + y)z

From (i) and (ii)
xy + z = xyz = (x + y) z