Proof:

We are given that angle A and angle B are acute angles such that cos A = cos B. We need to prove that angle A = angle B.

Step 1: Introduction

Let's assume that angle A and angle B are acute angles and cos A = cos B.

Step 2: Definition of Cosine

The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. In other words, cos A = adjacent side / hypotenuse and cos B = adjacent side / hypotenuse.

Step 3: Assume the triangles

Let's assume two right triangles with acute angles A and B. In both triangles, the adjacent side and the hypotenuse are equal.

Triangle 1:
Angle A
Adjacent side = x
Hypotenuse = y

Triangle 2:
Angle B
Adjacent side = x
Hypotenuse = y

Step 4: Substitute values

Since cos A = cos B, we can equate the ratios of the adjacent side to the hypotenuse in both triangles:

x / y = x / y

Step 5: Cross-multiplication

By cross-multiplication, we get:

xy = xy

Step 6: Simplification

The equation xy = xy implies that the product of the adjacent side and the hypotenuse is the same for both triangles.

Step 7: Conclusion

From the equation xy = xy, we can conclude that the adjacent side and the hypotenuse are equal for both triangles.

Since angle A and angle B are acute angles in the triangles, they have the same adjacent side and hypotenuse. Therefore, angle A = angle B.

Step 8: Summary

To summarize, we have shown that if angle A and angle B are acute angles such that cos A = cos B, then angle A = angle B. This is because the adjacent side and the hypotenuse are equal in both triangles formed by the given angles, leading to the equality of the angles.