The least positive integral value of k for which cos ⁡ 2 π 7 - sin ⁡ 2 π 7 sin ⁡ 2 π 7 cos ⁡ 2 π 7 k = 1 0 0 1 is

Step-by-step solution

A = cos ⁡ 2 π 7 - sin ⁡ 2 π 7 sin ⁡ 2 π 7 cos ⁡ 2 π 7 A 2 = cos ⁡ 2 π 7 - sin ⁡ 2 π 7 sin ⁡ 2 π 7 cos ⁡ 2 π 7 cos ⁡ 2 π 7 - sin ⁡ 2 π 7 sin ⁡ 2 π 7 cos ⁡ 2 π 7 = cos ⁡ 4 π 7 - sin ⁡ 4 π 7 sin ⁡ 4 π 7 cos ⁡ 4 π 7 Similarly, A k = cos ⁡ 2 k π 7 - sin ⁡ 2 k π 7 sin ⁡ 2 k π 7 cos ⁡ 2 k π 7 = 1 0 0 1 ⇒ cos ⁡ 2 k π 7 = 1 , sin ⁡ 2 k π 7 = 0 ⇒ 2 k π 7 = 2 n π , 2 k π 7 = m π ⇒ k = 7 n , k = 7 m 2 ⇒ Least positive integral value of k = 7