Question 596218
{{{(sin(u-v))/(sin(u)*cos(v)) = 1-cot(u)*tan(v)}}}
There are a number of things to consider when trying to figure out these identities:<ul><li>Match arguments -- Use argument-changing Trig properties (2x, 1/2x, A+B, A-B) to change arguments on the left to match those on the right</li><li>Match the number of terms -- Use algebra and/or Trig properties to get the same number of terms on the left side as there are on the right.</li><li>Match functions -- Use Trig properties to change functions on the left to match those on the right.</li><li>If none of the above look like they are going to work, try using Trig properties to change sec, csc, tan or cot on the left into sin and/or cos.</li></ul>Unfortunately, I cannot give a recipe of what to do. You just have to know your properties and algebra well enough to see which of the above looks promising. (If nothing looks promising, change everything into sin's and/or cos's and look again.)<br>
With your identity I see an argument of (u-v) on the left and no such argument on the left. So at some point we need to change that argument. That is where we will start. Using the sin(A-B) formula on sin(u-v) we get:
{{{(sin(u)cos(v)-cos(u)sin(v))/(sin(u)*cos(v)) = 1-cot(u)*tan(v)}}}<br>
The arguments now match. They are all u's and v's. Next we will match terms. The right side has two terms and the left side has one (big) term. We can use algebra to "un-subtract" the fraction:
{{{(sin(u)cos(v))/(sin(u)*cos(v))-(cos(u)sin(v))/(sin(u)*cos(v)) = 1-cot(u)*tan(v)}}}<br>
At this point you might see how close we are. The first fraction will simplify to a 1 which is what we want:
{{{1 - (cos(u)sin(v))/(sin(u)*cos(v)) = 1-cot(u)*tan(v)}}}
and we can "un-multiply" the second fraction to get the cot and tan we need:
{{{1 - (cos(u)/sin(u))(sin(v)/cos(v)) = 1-cot(u)*tan(v)}}}
{{{1 - cot(u)*tan(v) = 1-cot(u)*tan(v)}}}