Question

If $\angle A$ and $\angle B$ are acute angles such that cos A = cos B, then show that $\angle A=\angle B$

Answer 1

We know, $\cos \theta =\frac{AdjacentSide}{Hypotenuse}$

From the figure,

$\cos A=\frac{AC}{AB}$

$\cos B=\frac{BC}{AB}$

Given

$\cos A=\cos B$

$\frac{AC}{AB}=\frac{BC}{AB}$

Multiply both sides by $AB$

$\frac{AC}{AB}\times AB=\frac{BC}{AB}\times AB$

$AC=BC$

In Triangle ABC, $AC=BC$. Hence , $ABC$ is a scalene triangle.

If two sides of a triangle are equal, then the angles opposite to the sides are always equal.

Therefore, $\angle A=\angle B$