The number of solution of the equation ∑ r = 1 5 cos ⁡ r x = 0 lying in 0 , π is

Step-by-step solution

cos ⁡ x + cos ⁡ 2 x + cos ⁡ 3 x + cos ⁡ 4 x + cos ⁡ 5 x = 0 ⇒ sin ⁡ 5 x 2 sin ⁡ x 2 cos ⁡ 3 x = 0 ⇒ sin ⁡ 5 x 2 = 0 or cos ⁡ 3 x = 0 but sin ⁡ x 2 ≠ 0 If x ∈ 0 , π ⇒ 5 x 2 ∈ 0 , 5 π 2 ⇒ Two solutions 5 x 2 = π , 2 π If x ∈ 0 , π ⇒ 3 x ∈ 0 , 3 π ⇒ 3 solutions 3 x = π 2 , 3 π 2 , 5 π 2 So, total five solutions