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6. If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠ A = ∠ B.

6. If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠ A = ∠ B.

Grade:12th pass

1 Answers

Pawan Prajapati
askIITians Faculty 8741 Points
3 months ago
Let us assume the triangle ABC in which CD⊥AB Give that the angles A and B are acute angles, such that Cos (A) = cos (B) As per the angles taken, the cos ratio is written as AD/AC = BD/BC Now, interchange the terms, we get AD/BD = AC/BC Let take a constant value AD/BD = AC/BC = k Now consider the equation as AD = k BD …(1) AC = k BC …(2) By applying Pythagoras theorem in △CAD and △CBD we get, CD2 = BC2 – BD2 … (3) CD2 =AC2 −AD2 ….(4) From the equations (3) and (4) we get, AC2−AD2 = BC2−BD2 Now substitute the equations (1) and (2) in (3) and (4) K2(BC2−BD2)=(BC2−BD2) k2=1 Putting this value in equation, we obtain AC = BC ∠A=∠B (Angles opposite to equal side are equal-isosceles triangle)

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