If cos ,alpha + cos ,beta = | vedclass

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Question

Let $\cos \alpha+\cos \beta=\frac{3}{2}$

$\Rightarrow 2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}=\frac{3}{2}$ ..... $(i)$

and $\s

Vedclass · Accepted Answer

$\frac{7}{5}$

Vedclass · Answer

Let $\cos \alpha+\cos \beta=\frac{3}{2}$

$\Rightarrow 2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}=\frac{3}{2}$ ..... $(i)$

and $\sin \alpha+\sin \beta=\frac{1}{2}$

$\Rightarrow 2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}=\frac{1}{2}$ ..... $(ii)$

On dividing $(ii)$ by $( i)$, we get

$	an \left(\frac{\alpha+\beta}{2}ight)=\frac{1}{3}$

Given $: 	heta=\frac{\alpha+\beta}{2} \Rightarrow 2 	heta=\alpha+\beta$

Consider $\sin 2 	heta+\cos 2 	heta=\sin (\alpha+\beta)+\cos$
$(\alpha+\beta)$

$=\frac{\frac{2}{3}}{1+\frac{1}{9}}+\frac{1-\frac{1}{9}}{1+\frac{1}{9}}=\frac{6}{10}+\frac{8}{10}=\frac{7}{5}$

If $\cos \,\alpha + \cos \,\beta = \frac{3}{2}$ and $\sin \,\alpha + \sin \,\beta = \frac{1}{2}$ and $\theta $ is the the arithmetic mean of $\alpha $ and $\beta $ , then $\sin \,2\theta + \cos \,2\theta $ is equal to

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