Explain why the equation tanx + cotx = 1 does not have real solutions.

First of all, we need to express tangent and cotangent in terms of sine and cosine using the identities: tanx = sinx / cosx and cotx = cosx / sinx. Substituting these expressions into the initial equation we get: sinx / cosx + cosx / sinx = 1. Summing the two terms on the right hand side we obtain: [(sinx)^2 + (cosx)^2] / (sinxcosx) = 1.
Now it is important to remember two fundamental trigonometry identities in order to simplify the right hand side further. These identities are: (sinx)^2 + (cosx)^2 = 1 and sin2x = 2sinxcosx, which can be rearranged into sinxcosx = (1/2)*sin2x. Substituting these expressions in the numerator and denominator respectively, we get: 1/[(1/2)*sin2x] = 1. Rearranging: sin2x = 2. However, we know that the values of sine are between -1 and 1, hence there is no real value of x such that the equation is verified.