Question 240783
{{{cos(u-v)/(cos(u)sin(v))= tan(u)+cot(v)}}}
Since there is no (u-v) on the right side, we will use {{{cos(A-B) = cos(A)cos(B) + sin(A)sin(B)}}} on the cos(u-v) giving us:
{{{(cos(u)cos(v) + sin(u)sin(v))/(cos(u)sin(v))= tan(u)+cot(v)}}}
Since there are two terms on the right side, we will split the fraction on the left into two terms. (Think of it as "unadding".):
{{{(cos(u)cos(v))/(cos(u)sin(v)) + (sin(u)sin(v))/(cos(u)sin(v))= tan(u)+cot(v)}}}
As you can see, we can do some canceling in each fraction:
{{{(cross(cos(u))cos(v))/(cross(cos(u))sin(v)) + (sin(u)cross(sin(v)))/(cos(u)cross(sin(v)))= tan(u)+cot(v)}}}
leaving:
{{{cos(v)/sin(v) + sin(u)/cos(u)= tan(u)+cot(v)}}}
Since cos(v)/sin(v) = cot(v) and sin(u)/cos(u) = tan(u) we can substitute and get:
{{{cot(v) + tan(u) = tan(u)+cot(v)}}}
And, using the Commutative Property of Addition, we can change the order on the left to:
{{{tan(u)+ cot(v) = tan(u)+cot(v)}}}
And we are done.