Verify the identity. cotangent of x divided by quantity one plus cosecant of x equals quantity cosecant of x minus one divided by cotangent of x

We are asked to verify the function: cot x / (1+cscx) = (cscx-1) / cotx that is to simplify each side of the equation to be equal to each other. In this case, we can use the trigonometric identities to determine if both sides are equal to each other.

cot x = 1/tan x

tan x = sinx / cosx

Ergo, cot x = cos x / sinx

cscx = 1/sinx

left side: cotx / (1 + csc x) = cos x / sinx / (1+1/sinx) = cosx / (sinx + 1) * sinx/sinx = cosx / (sinx + 1)

Right side: (cscx-1) / cotx = ((1/sinx) - 1) * sinx/cosx = (1-sinx) * sinx/sinx*cosx =

1-sinx / cosx

using a calculator, these two are equal.