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Question

$$\cos \,A = \dfrac{3}{4}\Rightarrow 32\,\sin \left(\dfrac{A}{2}\right) \sin\left(\dfrac{5A}{2}\right)=$$


A
7
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B
8
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C
13
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D
11
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Solution

The correct option is A 7
Given, $$ \cos A= \dfrac{3}{4}$$
To Solve for  ,
$$32\sin\left ( \dfrac{A}{2} \right )\sin\left ( \dfrac{5A}{2} \right )$$
$$16\times 2\sin\left ( \dfrac{A}{2} \right )\sin\left ( \dfrac{5A}{2} \right )$$
$$16\left [ \cos\left ( \dfrac{A}{2} - \dfrac{5A}{2} \right ) -\cos\left ( \dfrac{A}{2} + \dfrac{5A}{2} \right ) \right ]$$
$$16\left [ \cos\left ( -2A \right ) -\cos\left ( 3A \right ) \right ]$$
$$16\left [ 1-2\cos^{2}A - 4\cos^{3}A + 3\cos A \right ]$$
Now, as we have
$$ \cos A = \dfrac{3}{4}$$
So, $$16\left [ 1-2\left ( \dfrac{3}{4} \right )^{2} - 4\left ( \dfrac{3}{4} \right )^{3} + 3\left ( \dfrac{3}{4} \right ) \right ]$$
$$ = 16\times \left [ \dfrac{7}{16} \right ]$$      
$$ =  7 $$

Mathematics

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