Find the real values of θ for which is purely real.
Since is purely real
Firstly, we need to solve the given equation and then take the imaginary part as 0
We rationalize the above by multiply and divide by the conjugate of (1 -2i cos θ)
We know that,
(a – b)(a + b) = (a2 – b2)
[∵ i2 = -1]
Since is purely real [given]
Hence, imaginary part is equal to 0
i.e.
⇒ 3 cos θ = 0 × (1 + 4 cos2θ)
⇒ 3 cos θ = 0
⇒ cos θ = 0
⇒ cos θ = cos 0
Since, cos θ = cos y
Then where n Є Z
Putting y = 0
where n Є Z
Hence, for is purely real.