Verified answer

Sorry I am not only using the left side

first cross multiply to clear the fractions (we hate to deal with them if we don't have to)

(cscθ - 1) / (cotθ) = (cotθ) / (csc + 1)

(cscθ - 1) (cscθ + 1) = cot^2θ

FOIL the left side

csc^θ - 1 = cot^2θ

but we know that from the pythagorean identity cot^2θ + 1 = csc^2θ

so the left side = cot^2θ

cot^2θ = cot^2θ

LHS : (cosecθ -1) /cotθ.

Mutiply and divide by (cosecθ + 1),

(cosecθ -1)(cosecθ+1) / cotθ (cosecθ+1)

(Cosec^2)θ -1 /cotθ (Cosecθ +1)

(cot^2)θ/cotθ (Cosecθ +1)

cotθ/Cosecθ +1 =RHS