\frac{1}{2}\left(1+\arctan \left(x\right)\right)=\frac{1+\sin \left(2x\right)}{1-\cos \left(2x\right)+\sin \left(2x\right)}
You asked:
Solve the equation \(\frac{1}{2} \cdot \left(1 + \arctan\left( x \right)\right) = \frac{1 + \sin\left( 2 x \right)}{1 - \cos\left( 2 x \right) + \sin\left( 2 x \right)}\) for the variable \(x\).
MathBot Answer:
The complex solutions are: \[\left\{x\; \middle|\; x \in \mathbb{R} \wedge \left(\sin{\left(2 x \right)} - \cos{\left(2 x \right)} + 1\right) \arctan{\left(x \right)} - \sin{\left(2 x \right)} - \cos{\left(2 x \right)} - 1 = 0 \right\} \setminus \left(\left\{n \pi\; \middle|\; n \in \mathbb{Z}\right\} \cup \left\{\frac{\pi \left(4 n + 3\right)}{4}\; \middle|\; n \in \mathbb{Z}\right\}\right)\]