(C3) Show that 4csc^2(x) - cot^2(x) = k can be expressed as sec^2(x) = (k-1)/(k-4) where k != 4

The student can answer this in several ways. One using the simple, known identities csc= 1/sin, cot=1/tan, sec=1/cos, tan=sin/cos, sin^2 + cos^2 = 1 and basic algebra is the following:
4csc^2 - cot^2 = k4/sin^2 - 1/tan^2 = k Substitute inverse fomulae4/sin^2 - cos^2/sin^2 = k Substitute tan4 - cos^2 = ksin^2 4 - cos^2 = k(1-cos^2) Write in terms of Coskcos^2 - cos^2 = k - 4cos^2(k - 1) = k-4 Gather Cos terms(k-1)/(k-4) = sec^2 Write in terms of inverses