If the number of principal solutions of the equation tan ⁡ 7 π cos ⁡ x = cot ⁡ 7 π sin ⁡ x is k , then k 5 is equal to

Step-by-step solution

tan ⁡ 7 π cos ⁡ x = tan ⁡ π 2 - 7 π sin ⁡ x ⇒ 7 π cos ⁡ x = n π + π 2 - 7 π sin ⁡ x ⇒ 7 π ( sin ⁡ x + cos ⁡ x ) = n + 1 2 π ⇒ sin ⁡ x + cos ⁡ x = 2 n + 1 14 ⇒ sin ⁡ x + π 4 = 2 n + 1 14 2 - 1 ≤ sin ⁡ x + π 4 ≤ 1 - 1 ≤ 2 n + 1 14 2 ⁡ ≤ 1 - 14 2 - 1 2 ⁡ ≤ n ≤ 14 2 - 1 2 - 10.4 ≤ n ≤ 9.4 ⇒ n = - 10 , - 9