f an angle B is complement of an angle A, what are the greatest and least values of cosAcosB respectively?0,−1212,−11,012,−12

The greatest and least values of cosAcosB respectively are 0 and -1/2.

Here's why:

Two angles are complementary if the sum of their measures is 90 degrees. So, if angle B is the complement of angle A, then A + B = 90 degrees.

The cosine of an angle is the ratio of the adjacent side to the hypotenuse in a right triangle. The product of the cosines of two angles is given by cosAcosB.

The maximum value of cosAcosB occurs when both cosA and cosB are at their maximum, which is 1. However, since A and B are complementary, when one is at its maximum, the other is at its minimum (0). So, the maximum value of cosAcosB is 0.

The minimum value of cosAcosB occurs when one of the cosines is at its maximum (1) and the other is at its minimum (-1). However, since A and B are complementary, the minimum value of one of the cosines is 0, not -1. So, the minimum value of cosAcosB is -1/2. This occurs when A = B = 45 degrees, because cos(45) = sqrt(2)/2, and (sqrt(2)/2)^2 = 1/2.