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Question
Mark the correct alternative in each of the following:
In any ∆ABC, the value of \[2ac\sin\left( \frac{A - B + C}{2} \right)\] is
Options
\[a^2 + b^2 - c^2\]
\[c^2 + a^2 - b^2\]
\[b^2 - c^2 - a^2\]
\[c^2 - a^2 - b^2\]
Solution
In ∆ABC,
\[A + B + C = \pi \left( \text{ Angle sum property } \right)\]
\[ \Rightarrow A + C = \pi - B\]
\[\therefore 2ac\sin\left( \frac{A - B + C}{2} \right)\]
\[ = 2ac\sin\left( \frac{\pi - 2B}{2} \right)\]
\[ = 2ac\sin\left( \frac{\pi}{2} - B \right)\]
\[ = 2ac\cos B\]
\[= 2ac\left( \frac{c^2 + a^2 - b^2}{2ca} \right) \left( \text{ Using cosine rule } \right)\]
\[ = c^2 + a^2 - b^2\]
Hence, the correct answer is option (b).
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