Question

6. If Z A and B are acute angles such that cos A = cos B, then show that ZA= Z B.
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Answer 1

We are given that,

$\qquad \rm {• cos \: A = cos \: B }$

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We have to show,

$\qquad \rm {• ∠A = ∠B }$

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Construction,

$\dag$ Draw a triangle ABC where C = 90°.

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Solution

We know,

$\qquad \rm {• cos \: A = cos \: B }$

$\implies { \rm \dfrac{B}{H} = \dfrac{B}{H} }$

$\implies { \rm \dfrac{AC}{AB} = \dfrac{BC}{AB} }$

$\implies \rm{ AC \times \cancel {AB} = BC \times \cancel {AB} }$

$\rm {\implies AC = BC }$

Now , we know that if two sides of triangle are equal than it is isosceles triangle and we also know that in isosceles triangle, if two opposite sides are equal than their angle will also be equal.

$\implies { \rm \angle A= \angle B }$

Thus, prooved!