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Transformation Formulae — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTransformation Formulae1 Mark
Question
If $\sin\text{A}+\sin\text{B}=\alpha\text{ and }\cos\text{A}+\cos\text{B}=\beta,$ than write the value of $\tan\Big(\frac{\text{A+B}}{2}\Big).$
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Answer

We have, $\sin\text{A}+\sin\text{B}=\alpha...(\text{i})$ $\text{and },\cos\text{A}+\cos\text{B}=\beta...(\text{ii})$ Now, $\frac{\sin\text{A}+\sin\text{B}}{\cos\text{A}+\cos\text{B}}=\frac{\alpha}{\beta}$ $\Rightarrow\ \frac{2\sin\Big(\frac{\text{A+B}}{2}\Big)\cos\Big(\frac{\text{A}-\text{B}}{2}\Big)}{2\cos\Big(\frac{\text{A+B}}{2}\Big)\cos\Big(\frac{\text{A}-\text{B}}{2}\Big)}=\frac{\alpha}{\beta}$$\Rightarrow\ \frac{\sin\Big(\frac{\text{A+B}}{2}\Big)}{\cos\Big(\frac{A+B}{2}\Big)}=\frac{\alpha}{\beta}$
$\Rightarrow\ \tan\Big(\frac{\text{A+B}}{2}\Big)=\frac{\alpha}{\beta}$

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