Question

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




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Answer 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)