Prove that, $cos 2 A + cos 2 B - cos 2 C = 1 - 4 sin A sin B cos C$ .

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Solution

Here we want $+ 1$ and as such we write $cos 2 A = 1 - 2 {sin}^{2} A$ and combine the other two terms.
L.H.S. $= 1 - 2 {sin}^{2} A + 2 sin (B + C) sin (C - B)$ (Note)
$= 1 - 2 {sin}^{2} A - 2 sin A sin (B - C)$ $[∵ sin (- θ) = - sin θ]$
$= 1 - 2 sin A [sin A + sin (B - C)]$
$= 1 - 2 sin A [sin (B + C) + sin (B - C)]$
$= 1 - 2 sin A (2 sin B cos C)$
$= 1 - 4 sin A sin B cos C$ .

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