# 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.

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