$s i n^{2} A = c o s^{2} (A + B) + c o s^{2} B - 2 c o s (A - B) c o s A c o s B$

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Solution

$\begin{matrix} G i v e n : s i n^{2} A = c o s^{2} (A + B) + c o s^{2} B - 2 c o s (A - B) c o s A c o s B R . H . S = c o s^{2} (A + B) + c o s^{2} B - 2 c o s (A - B) c o s A c o s B [\begin{matrix} u s i n g i d e n t i t y R u l e s, c o s^{2} (A \pm B) = cos A cos B - sin A sin B \end{matrix} \Rightarrow (c o s A c o s B - s i n A s i n B)^{2} + c o s^{2} B - 2 (cos A cos B - sin A sin B) c o s A c o s B \Rightarrow {cos}^{2} A {cos}^{2} B + {sin}^{2} A {sin}^{2} B - 2 c o s A cos B . sin A sin B + c o s^{2} B - 2 (cos A cos B - sin A sin B) c o s A c o s B \Rightarrow {cos}^{2} A {cos}^{2} B + {sin}^{2} A {sin}^{2} B - 2 c o s A cos B . sin A sin B + c o s^{2} B - 2 {cos}^{2} A {cos}^{2} B + 2 sin A sin B . c o s A c o s B \Rightarrow {sin}^{2} A {sin}^{2} B + c o s^{2} B - {cos}^{2} A {cos}^{2} B \Rightarrow {sin}^{2} A {sin}^{2} B + c o s^{2} B (1 - c o s^{2} A) \Rightarrow {sin}^{2} A {sin}^{2} B + c o s^{2} B . {sin}^{2} A \Rightarrow {sin}^{2} A ({sin}^{2} B + c o s^{2} B) ∴ {sin}^{2} A = L . H . S H e r e L . H . S = R . H . S \end{matrix}$

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{cos}^{2} (A - B) + {cos}^{2} B - 2 cos (A - B) cos A cos B

m s i n^{2} A = c o s^{2} (A - B) + c o s^{2} B - 2 c o s (A - B) c o s A c o s B .

m

c o s^{2} (A - B) + c o s^{2} B - 2 c o s (A - B) c o s A c o s B

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\frac{\sin (A - B)}{\cos A \cos B} + \frac{\sin (B - C)}{\cos B \cos C} + \frac{\sin (C - A)}{\cos C \cos A} = 0

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