For the domain of definition of the given equation, we have
(i) 2cos2x−1≠0⇒x≠nπ±π6
(ii) tanx≠0⇒x≠±nπ2 [ For odd multiples of π2,tanx is not defined ]
(iii) cos2x−3sin2x≠0⇒x≠nπ±π6
Also, 2cos2x−1=2(cos2x−sin2x)−(cos2x+sin2x)=cos2x−3sin2x
Now, the given equation reduces to
bsinx=b+sinx⇒sinx=bb−1
∵−1≤sinx≤1
∴−1≤bb−1≤1
⇒bb−1+1≤0 and bb−1−1≤0
⇒2b−1b−1≥0 and 1b−1≤0
⇒b≤12⇒b>1 and b<1⇒b≤12
when b=12⇒sinx=1, which is not possible
∴b<12