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Transformation Formulae — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTransformation Formulae4 Marks
Question
Prove that:
$\text{If}\cos\text{A}+\cos\text{B}=\frac{1}{2}\text{ and }\sin\text{A}+\sin\text{B}=\frac{1}{4},$ prove that $\tan\Big(\frac{\text{A+B}}{2}\Big)=\frac{1}{2}.$
✓

Answer

We have,
$\text{LHS}=\cos\text{A}+\cos\text{B}=\frac{1}{2}$
$\text{and},\ \sin\text{A}+\sin\text{B}=\frac{1}{4}$
Now,
$\frac{\sin\text{A}+\sin\text{B}}{\cos\text{A}+\cos\text{B}}=\frac{\frac{1}{4}}{\frac{1}{2}}$ [On dividing]
$\Rightarrow\ \frac{2\sin\Big(\frac{\text{A}+\text{B}}{2}\Big)\cos\Big(\frac{\text{A}-\text{B}}{2}\Big)}{2\cos\Big(\frac{\text{A}+\text{B}}{2}\Big)\cos\Big(\frac{\text{A}-\text{B}}{2}\Big)}=\frac{1}{2}$
$\Rightarrow\ \frac{\sin\Big(\frac{\text{A}+\text{B}}{2}\Big)}{\cos\Big(\frac{\text{A}+\text{B}}{2}\Big)}=\frac{1}{2}$
$\Rightarrow\ \tan\Big(\frac{\text{A}+\text{B}}{2}\Big)=\frac{1}{2}$
$\Rightarrow\ \text{RHS}$
Hence proved.

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