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

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Today  i am solving introduction to trigonometry question of exercise 8.1 of question no. 6 but i can’t solve this question .Find the best way to solve this question in a easy way  its so important question for this chapter.If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠ A = ∠ B.

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  1. 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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