Question

Use rolle's theorem to show that between any two roots of (x-1)sinx=0 there exist at least one root of tanx=1-x

Answer 1

As I showed before,

If y = (x-1) sinx, then
y' = sinx + (x-1)cosx = k sin(x+θ)
where cosθ = 1/√(1+(x-1)^2)
since k is never zero, the zeroes of y' are the same as the zeroes of y, just shifted by θ. That is, they are between the zeroes of y