Question 569237
changing everything on left to sin or cos and FOILing


{{{((cos(A)/sin(A)) + ((cos(A)*sin(A)) / sin(A)) - cos(A) - (sin(A)*cos(A))) / (cos(A))^3 = csc(A)}}}


cancelling cos(A) on the left


{{{((1/sin(A)) + (sin(A) / sin(A)) - 1 - sin(A)) / (cos(A))^2 = csc(A)}}} ___ {{{((1/sin(A)) + 1 - 1 - sin(A)) / (cos(A))^2 = csc(A)}}}


{{{((1/sin(A)) - sin(A)) / (cos(A))^2 = csc(A)}}}


multiplying left side by [sin(A) / sin(A)] ___ {{{(1 - (sin(A))^2) / (sin(A)*(cos(A))^2) = csc(A)}}} ___ {{{(cos(A))^2 / (sin(A)*(cos(A))^2) = csc(A)}}}


canceling cos^2(A) on the left side ___ {{{1 / sin(A) = csc(A)}}}