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If Cos a + Cos B = 1 2 and Sin a + Sin B = 1 4 , Prove that Tan ( a + B 2 ) = 1 2

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Question

\[\text{ If } \cos A + \cos B = \frac{1}{2}\text{ and }\sin A + \sin B = \frac{1}{4},\text{ prove that }\tan\left( \frac{A + B}{2} \right) = \frac{1}{2} .\]

 

Sum
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Solution

Given:
sin A + sin B = \[\frac{1}{4}\]         .....(i)
cos A + cos B = \[\frac{1}{2}\]         .....(ii)
Dividing (i) by (ii):
\[\Rightarrow \frac{\sin A + \sin B}{\cos A + \cos B} = \frac{\frac{1}{4}}{\frac{1}{2}}\]
\[ \Rightarrow \frac{2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)}{2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)} = \frac{1}{2} \left[ \because \sin A + \sin B = 2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)\text{ and }\cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right]\]
\[ \Rightarrow \frac{\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)}{\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)} = \frac{1}{2}\]
\[ \Rightarrow \tan\left( \frac{A + B}{2} \right)=\frac{1}{2}\]
Hence proved.

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Transformation Formulae
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Chapter 8: Transformation formulae - Exercise 8.2 [Page 19]

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R.D. Sharma Mathematics [English] Class 11
Chapter 8 Transformation formulae
Exercise 8.2 | Q 10 | Page 19

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