Question:
If $\angle A$ and $\angle B$ are acute angles such that $\cos A=\cos B$, then show that $\angle A=\angle B$.
Solution:
Given:
$\cos A=\cos B$.....(1)
To show: $\angle A=\angle B$
$\triangle A B C$ is as shown in figure below
Therefore
$\frac{A C}{A B}=\frac{B C}{A B}$
Now observe that denominator of above equality is same that is AB
Hence $\frac{A C}{A B}=\frac{B C}{A B}$ only when $A C=B C$
Therefore $A C=B C$.....(2)
We know that when two sides of a triangle are equal, then angle opposite to the sides are also equal.
Therefore from equation (2)
We can say that
Angle opposite to side AC = Angle opposite to side BC
Therefore,
$\angle B=\angle A$
Hence, $\angle A=\angle B$