If $\tan(\frac{\pi}{4}+\frac{\alpha}{2}) = \tan^3(\frac{\pi}{4}+\frac{\beta}{2})$, then $\frac{3+\sin^2\beta}{1+3\sin^2\beta}=$
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The identity $\tan^2(\frac{\pi}{4}+\frac{\theta}{2}) = \frac{1+\sin\theta}{1-\sin\theta}$ is a very useful tool for converting between tangent of a shifted half-angle and the sine of the full angle. It simplifies many trigonometric problems involving such expressions.