Three positive acute angles α ,   β and γ satisfy the relation tan ⁡ β 2 = 1 3 cot ⁡ α 2 and cot ⁡ γ 2 = 1 2 3 tan ⁡ α 2 + cot ⁡ α 2 . Then, the value of α + β + γ is equal to

Step-by-step solution

We have, tan ⁡ α 2 tan ⁡ β 2 = 1 3 ⇒ 1 - tan ⁡ α 2 tan ⁡ β 2 = 2 3 Now, tan ⁡ α + β 2 = tan ⁡ α 2 + tan ⁡ β 2 1 - tan ⁡ α 2 tan ⁡ β 2 = tan ⁡ α 2 + tan ⁡ β 2 2 3 ⇒ tan ⁡ α + β 2 = 3 2 tan ⁡ α 2 + tan ⁡ β 2 = 3 2 tan ⁡ α 2 + 1 2 3 tan ⁡ β 2 = 3 2 tan ⁡ α 2 + 1 2 cot ⁡ α 2 = cot ⁡ γ 2 ⇒ tan ⁡ α + β 2 = cot ⁡ γ 2 = 1 tan ⁡ γ 2 ⇒ 1 - tan ⁡ α + β 2 ⋅ tan ⁡ γ 2 = 0 ⇒ α + β 2 + γ 2 = π 2 ⇒ α + β + γ = π