$\dfrac{1}{1 + \sin^2\theta} + \dfrac{1}{1 + \operatorname{cosec}^2\theta} = ?$

Since $\operatorname{cosec}^2\theta = \dfrac{1}{\sin^2\theta}$, the second denominator becomes $1 + \dfrac{1}{\sin^2\theta} = \dfrac{1+\sin^2\theta}{\sin^2\theta}$, so the second fraction equals $\dfrac{\sin^2\theta}{1+\sin^2\theta}$. Adding gives $\dfrac{1 + \sin^2\theta}{1+\sin^2\theta} = 1$.