If the expression ( 1 + tan ⁡ x + tan 2 ⁡ x ) ( 1 - cot ⁡ x + cot 2 ⁡ x ) is positive, then the complete set of values of x is

Question

If the expression ( 1 + tan ⁡ x + tan 2 ⁡ x ) ( 1 - cot ⁡ x + cot 2 ⁡ x ) is positive, then the complete set of values of x is, 0 , π 2, 0 , π, R - x = n π 2 , n ∈ I, 0 , ∞

MARKS App · Accepted Answer

(1 + tan ⁡ x + tan 2 ⁡ x ) ( 1 + t a n 2 x - tan ⁡ x ) tan 2 ⁡ x > 0 ⇒ 1 + tan 2 ⁡ x 2 - tan 2 ⁡ x tan 2 ⁡ x > 0 Since, 1 + tan 2 ⁡ x > tan 2 ⁡ x , ∀ x ∈ R - x = n π 2 , n ∈ I Hence, given expression is positive for all values of x ∈ R - x = n π 2 , n ∈ I

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