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m - n Theorem

In Δ ABC, the...

Question

In $Δ A B C,$ the median AD divides $∠ B A C$ such that $∠ B A D : ∠ C A D = 2 : 1.$ then $c o s \frac{A}{3}$ is equal to

$\frac{s i n B}{2 s i n C}$

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$\frac{s i n C}{2 s i n B}$

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$\frac{2 s i n B}{s i n C}$

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None of these

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Solution

The correct option is A
$\frac{s i n B}{2 s i n C}$

$\frac{A}{3} = ∠ C A D = θ N o w, (1 + 1) c o t α = 1. c o s 2 θ - 1. c o t θ \Rightarrow 2 c o t (B + 2 θ) = c o t 2 θ - c o t θ \Rightarrow c o t (B + 2 θ) + c o t θ = c o t 2 θ - c o t (B + 2 θ) \frac{s i n (B + 3 θ)}{s i n (B + 2 θ) . s i n θ} = \frac{s i n B}{s i n (B + 2 θ) . s i n 2 θ} \Rightarrow \frac{s i n (B + A)}{s i n θ} = \frac{s i n B}{s i n 2 θ} \Rightarrow s i n C = \frac{s i n B}{2 c o s θ} \Rightarrow c o s \frac{A}{3} = \frac{s i n B}{2 s i n C}$

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