Answer each of the following questions in one word or one sentence or as per exact requirement of for question:

$I n a Δ A B C, i f c o s A = \frac{s i n B}{2 s i n C}, then show that c = a.$

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Solution

$L e t \frac{s i n A}{a} = \frac{s i n B}{b} = \frac{s i n C}{c} = k c o s A = \frac{s i n B}{2 s i n C} \Rightarrow (\frac{b^{2} + c^{2} - a^{2}}{2 b c}) = \frac{k b}{2 k c} \Rightarrow (\frac{b^{2} + c^{2} - a^{2}}{b}) = b \Rightarrow b^{2} + c^{2} - a^{2} = b^{2} \Rightarrow c^{2} = a^{2} \Rightarrow c = a$

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