$\tan ^{-1} \frac{2 \sin \frac{\alpha}{2} \sin \left(\frac{\pi}{4}-\frac{\beta}{2}\right) \cos \frac{\alpha}{2} \cos \left(\frac{\pi}{4}-\frac{\beta}{2}\right)}{\cos ^{2} \frac{\alpha}{2} \cos ^{2}\left(\frac{\pi}{4}-\frac{\beta}{2}\right)-\sin ^{2} \frac{\alpha}{2} \sin ^{2}\left(\frac{\pi}{4}-\frac{\beta}{2}\right)}= ?$

Solve:$\begin{array}{l}=\tan ^{-1} \frac{2 \sin \frac{\alpha}{2} \sin \left(\frac{\pi}{4}-\frac{\beta}{2}\right) \cos \frac{\alpha}{2} \cos \left(\frac{\pi}{4}-\frac{\beta}{2}\right)}{\cos ^{2} \frac{\alpha}{2} \cos ^{2}\left(\frac{\pi}{4}-\frac{\beta}{2}\right)-\sin ^{2} \frac{\alpha}{2} \sin ^{2}\left(\frac{\pi}{4}-\frac{\beta}{2}\right)} \\\text { now, } 2 \sin \frac{\alpha}{2} \sin \left(\frac{\pi}{4}-\frac{\beta}{2}\right) \cos \frac{\alpha}{2} \cos \left(\frac{\pi}{4}-\frac{\beta}{2}\right) \\=\frac{1}{2} \sin \alpha \sin 2\left(\frac{\pi}{4}-\frac{\beta}{2}\right) \\=\frac{1}{2} \sin \alpha \sin \left(\frac{\pi}{2}-\beta\right) \\=\frac{1}{2} \sin \alpha \cos \beta\text { , and } \\\cos ^{2} \frac{\alpha}{2} \cos ^{2}\left(\frac{\pi}{4}-\frac{\beta}{2}\right)-\sin ^{2} \frac{\alpha}{2} \sin ^{2}\left(\frac{\pi}{4}-\frac{\beta}{2}\right) \\ \left.=\cos \frac{\alpha}{2} \cos \left(\frac{\pi}{4}-\frac{\beta}{2}\right)+\sin \frac{\alpha}{2} \sin \left(\frac{\pi}{4}-\frac{\beta}{2}\right)\right\} \\ \left\{\cos \frac{\alpha}{2} \cos \left(\frac{\pi}{4}-\frac{\beta}{2}\right)-\sin \frac{\alpha}{2} \sin \left(\frac{\pi}{4}-\frac{\beta}{2}\right)\right\} \\ =\cos \left(\frac{\alpha}{2}+\frac{\beta}{2}-\frac{\pi}{4}\right) \cdot \cos \left(\frac{\alpha}{2}-\frac{\beta}{2}+\frac{\pi}{4}\right) \\ =\cos \left\{\frac{\alpha}{2}+\left(\frac{\beta}{2}-\frac{\pi}{4}\right)\right\} \cdot \cos \left\{\frac{\alpha}{2}-\left(\frac{\beta}{2}-\frac{\pi}{4}\right)\right\} \\ =\frac{1}{2}\left\{\cos 2 \cdot \frac{\alpha}{2}+\cos 2 \cdot\left(\frac{\beta}{2}-\frac{\pi}{4}\right)\right\} \\ =\frac{1}{2}\left\{\cos \alpha+\cos \left(\beta-\frac{\pi}{2}\right)\right\} \\ =\frac{1}{2}\{\cos \alpha+\sin \beta) \\=\tan ^{-1} \frac{2 \sin \frac{\alpha}{2} \sin \left(\frac{\pi}{4}-\frac{\beta}{2}\right) \cos \frac{\alpha}{2} \cos \left(\frac{\pi}{4}-\frac{\beta}{2}\right)}{\cos ^{2} \frac{\alpha}{2} \cos ^{2}\left(\frac{\pi}{4}-\frac{\beta}{2}\right)-\sin ^{2} \frac{\alpha}{2} \sin ^{2}\left(\frac{\pi}{4}-\frac{\beta}{2}\right)} = tan^{-1} \frac{sin \alpha cos \beta}{cos \alpha + sin \beta}\end{array}$