$$ \begin{vmatrix} \sin \theta & -\cos \theta & \lambda+1 \\ \cos \theta & \sin \theta & -\lambda \\ \lambda & \lambda +1 & \cos \theta \\ \end{vmatrix} = 2(\cos \theta)( \lambda^2 + \lambda + 1) $$
Attempt: Dividing R1 and R2 by sin $\theta $ and cos $\theta $, and then doing R2 -> R2-R3 and R1-> R1-R3 But it would still be complicated to solve the equation.