Math Problem Solver

$$\sin^{2}{\left(x \right)} + \tan{\left(x \right)} = \sin^{2}{\left(\arctan{\left(\frac{1}{3} \right)} + \arctan{\left(\frac{1}{2} \right)} \right)} + \tan{\left(\arctan{\left(\frac{1}{3} \right)} + \arctan{\left(\frac{1}{2} \right)} \right)}$$

$$\begin{aligned}\operatorname{atan}{\left(\frac{1}{3} \right)} + \operatorname{atan}{\left(\frac{1}{2} \right)} &= x\\- x &= - \operatorname{atan}{\left(\frac{1}{2} \right)} - \operatorname{atan}{\left(\frac{1}{3} \right)}\\x &= \operatorname{atan}{\left(\frac{1}{3} \right)} + \operatorname{atan}{\left(\frac{1}{2} \right)}\end{aligned}$$

By substituting this value into the expression, we find that: $$\begin{aligned}\sin^{2}{\left(x \right)} + \tan{\left(x \right)} &= \sin^{2}{\left(\arctan{\left(\frac{1}{3} \right)} + \arctan{\left(\frac{1}{2} \right)} \right)} + \tan{\left(\arctan{\left(\frac{1}{3} \right)} + \arctan{\left(\frac{1}{2} \right)} \right)}\\&= \sin^{2}{\left(\arctan{\left(\frac{1}{3} \right)} + \arctan{\left(\frac{1}{2} \right)} \right)} + \tan{\left(\arctan{\left(\frac{1}{3} \right)} + \arctan{\left(\frac{1}{2} \right)} \right)}\end{aligned}$$