Reciprocal equations and restrictions

sec u=1/cos u, cosecu=1/sin u, and cot u=cos u/sin u. Each denominator must be nonzero. A reciprocal cannot be zero: 1/a=0 has no finite solution. But cot u=0 is possible when cos u=0 and sin u 0. 1. Record the original restrictions. 2. For a nonzero right side, invert secant or cosecant and solve the basic ratio equation. 3. Treat zero cotangent through its numerator. 4. Check every candidate in the original expression. Example: cot x=0, 0 <= x<2pi, gives x=pi/2,3pi/2. Both have nonzero sine; tangent is undefined there, so replacing cotangent by 1/tan x would conceal the solutions. Exam check: a fraction equals zero only when its numerator is zero and its denominator is defined and nonzero. Clearing a denominator does not make its excluded angles valid.