If the complex numbers sin ⁡ x + i cos ⁡ 2 x and cos ⁡ x - i sin ⁡ 2 x are conjugate of each other, then the number of values of x in the interval 0,2 π is equal to (where, i 2 = - 1 )

Step-by-step solution

sin ⁡ x = cos ⁡ x and cos ⁡ 2 x = sin ⁡ 2 x ⇒ sin ⁡ x = cos ⁡ x & 2 c o s 2 x - 1 = 2 sin ⁡ x cos ⁡ x ⇒ sin ⁡ x = cos ⁡ x & 2 c o s 2 x - 1 = 2 c o s 2 x ⇒ sin ⁡ x = cos ⁡ x & - 1 = 0 (Which is impossible) Hence, no value of x exists