Show taht `2tan^-1(tan(alpha/2)tan(pi/4-beta/2))=tan^-1((sinalphacosbeta)/(cosalpha+sinbeta))`

Explanation

L.H.S.

`=\tan ^{-1} \frac{2 \tan \frac{\alpha}{2} \cdot \tan (\frac{\pi}{4}-\frac{\beta}{2})}{1-\tan ^{2} \frac{\alpha}{2} \tan ^{2}(\frac{\pi}{4}-\frac{\beta}{2})}(\text { since } 2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^{2}})`



`\tan ^{-1} \frac{2 \tan \frac{\alpha}{2} \frac{1-\tan \frac{\beta}{2}}{1+\tan \frac{\beta}{2}}}{1-\tan ^{2} \frac{\alpha}{2}(\frac{1-\tan \frac{\beta}{2}}{1+\tan \frac{\beta}{2}})^{2}}`



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