# Calculate expression E=a*(b*cosC-c*cosB) if a,b,c are the lengths of the triangle ABC.

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### 1 Answer

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.**