In dimensional analysis, according to rule a physical quantity can be added or subtracted if and only if the physical quantities have same dimension.
For example, we can add 5 kg apple and 5 kg watermelon but cannot add 5 kg apple and 100 metre race because they have different dimensional unit.
Similarly, in the given question 1 has no unit and dimension. Therefore, θ should also not have any unit and dimension in order to be added with 1. Hence θ is a dimensionless quantity.
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