Find the sum of the smallest positive and largest negative Roots of the equation (in degrees) cos2x (tg2x + 1) = 0

The equality cos 2x (tg 2x + 1) = 0 is true in cases when cos 2x = 0, and in cases when tg 2x = -1.

Therefore, the equation cos 2x (tan 2x + 1) = 0 has two solutions:

First: 2x = 90 ° + 180 ° ⋅ n, where n is an integer. Hence: x = 45 ° + 90 ° ⋅ n, n ∈ Z.

Second: 2x = arctan (-1) + 180 ° ⋅ k = -45 ° + 180 ° ⋅ k, where k is an integer.

Hence: x = -22.5 ° + 90 ° ⋅ k, k ∈ Z.

Of the two solutions, the smallest positive root is x = 45 °, and the largest negative root is -22.5 °.

Their sum is 45 ° + (-22.5 °) = 45 ° – 22.5 ° = 22.5 °.

Answer: 22.5 °.