If 3 + cot ⁡ 80 o cot ⁡ 20 o cot ⁡ 80 o + cot ⁡ 20 o = t a n π k , then the value of k is (where, π k is an acute angle)

Question

If 3 + cot ⁡ 80 o cot ⁡ 20 o cot ⁡ 80 o + cot ⁡ 20 o = t a n π k , then the value of k is (where, π k is an acute angle),

MARKS App · Accepted Answer

3 sin ⁡ 80 o sin ⁡ 20 o + cos ⁡ 80 o cos ⁡ 20 o sin ⁡ 20 o cos ⁡ 80 o + cos ⁡ 20 o sin ⁡ 80 o = 2 sin ⁡ 20 o sin ⁡ 80 o + cos ⁡ 80 o cos ⁡ 20 o + sin ⁡ 80 o sin ⁡ 20 s i n 20 o + 80 o = cos ⁡ 60 o - cos ⁡ 100 o + c o s 80 o - 2 0 o sin ⁡ 100 o = 2 cos ⁡ 60 o - cos ⁡ 100 o sin ⁡ 100 o = 1 - cos ⁡ 100 o sin ⁡ 100 o = 2 s i n 2 5 0 ∘ 2 sin ⁡ 50 o cos ⁡ 50 o = tan ⁡ 50 o = t a n 5 π 18 = t a n π 18 5 ⇒ k = 18 5 = 3.6

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