Question

If sin A + sin B = α and cos A + cos B = β, then write the value of tan $\left(\frac{A+B}{2}\right)$.

Answer 1

Given:
sin A + sin B = α .....(i)
cos A + cos B = β .....(ii)
Dividing (i) by (ii):

$$\begin{gathered} \Rightarrow \frac{\sin A+\sin B}{\cos A+\cos B}=\frac{\alpha }{\beta } \\ \Rightarrow \frac{2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)}{2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)}=\frac{\alpha }{\beta }\left[\because \sin A+\sin B=2\sin \left(\frac{A+B}{2}\right)\cos \left({\displaystyle \frac{A-B}{2}}\right)\mathrm{and}\cos A+\cos B=2\cos \left(\frac{A+B}{2}\right)\cos \left({\displaystyle \frac{A-B}{2}}\right)\right] \\ \Rightarrow \frac{\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)}{\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)}=\frac{\alpha }{\beta } \\ \Rightarrow \tan \left(\frac{A+B}{2}\right)=\frac{\alpha }{\beta } \end{gathered}$$