Question

If $\angle A$ and $\angle B$ are acute angles such that $\cos \left(A\right)=\cos \left(B\right)$, then show that $\angle A=\angle B$.

Answer 1

Prove the given angles are equal based on the given information.

Given: $\angle A$ and $\angle B$ are acute angles such that $\cos \left(A\right)=\cos \left(B\right)$.

To prove: $\angle A=\angle B$

Proof:

Let us have right angle triangle $△ABC$

As we know $\cos$ is the ratio of the adjacent to the hypotenuse.
For angle $B$ , adjacent $=BC$ , hypotenuse $=AB$
So, $\cos \left(B\right)=\frac{BC}{AB}\dots \left(i\right)$
For angle $A$, adjacent $=AC$ , hypotenuse $=AB$
So,$\cos \left(A\right)=\frac{AC}{AB}\dots \left(ii\right)$
$\because \cos \left(A\right)=\cos \left(B\right)$ [given]

So from equations $\left(i\right)$ and $\left(ii\right)$
$\Rightarrow \frac{AC}{AB}=\frac{BC}{AB}$

$\Rightarrow AC=BC$
In a triangle, if two sides are equal then their opposite angles are also equal.
Therefore, $\mathrm{\angle }A=\mathrm{\angle }B$
Hence proved.