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Since the type of triangle is not indicated in the given enunciation, we'll consider an acute triangle.
We'll apply cosine theorem in an acute triangle, to express the terms cos C and cos B.
The lengths of the sides of the triangle are: BC = a, AC = b, AB = c.
cos C = (a^2 + b^2 - c^2)/2ab
cos B = (a^2 + c^2 - b^2)/2ac
We'll substitute cos C and cos B into the expression to be calculated.
E = a*[b*(a^2 + b^2 - c^2)/2ab - c*(a^2 + c^2 - b^2)/2ac]
We'll simplify and we'll get:
E = a*[(a^2 + b^2 - c^2)/2a - (a^2 + c^2 - b^2)/2a]
E = a^2/2 + b^2/2 - c^2/2 - a^2/2 - c^2/2 + b^2/2
We'll eliminate like terms and we'll combine the like terms:
E = 2b^2/2 - 2c^2/2
E = b^2 - c^2
The requested value of the expression is represented by the difference of the squares: E = b^2 - c^2.
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