If x = ∑ n = 0 ∞ - 1 n tan 2 ⁡ θ and y = ∑ n = 0 ∞ cos 2 n ⁡ θ , for 0 θ π 4 , then:

Question

If x = ∑ n = 0 ∞ - 1 n tan 2 ⁡ θ and y = ∑ n = 0 ∞ cos 2 n ⁡ θ , for 0 θ π 4 , then:, x 1 + y = 1, y 1 - x = 1, y 1 + x = 1, x 1 - y = 1

MARKS App · Accepted Answer

y = 1 + cos 2 ⁡ θ + cos 4 ⁡ θ + … Above series is an infinite G.P, apply sum of infinite G.P. ⇒ y = 1 1 - cos 2 ⁡ θ ⇒ 1 y = sin 2 θ x = 1 1 - - tan 2 ⁡ θ = 1 sec 2 ⁡ θ ⇒ cos 2 ⁡ θ = x ⇒ 1 y + x = 1 ⇒ y 1 - x = 1

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