```Question 604161
{{{tan(theta)/(sec(theta)-1) = csc(theta) + cot(theta)}}}
When trying to do these identities, one of the things to aim for is to match the number of terms on each side of the equation. Your equation has just one term on the left and it has two terms on the right. So somehow we need to expand the left side from one term to two.<br>
Changing the number of terms is often done by one of the following:<ul><li>Using a Trig property that has a different number of terms on the two sides. For example, {{{tan^2(x) + 1 = sec^2(x)}}}</li><li>"un-adding" or "un-subtracting" a fraction. For example: {{{(a+b)/c = a/c + b/c}}} or {{{(a-b)/c = a/c - b/c}}}</li></ul>
At first look, it doesn't seem that either of these will work on your equation. There are no Trig properties involving tan or sec that would seem to let us expand the left side into two terms. And the fraction has only one term so we can't "un-add" (or "un-substract")... at least not yet.<br>
If we multiply the numerator and denominator of the fraction by a two term expression, we will have the two terms we need in the numerator to "un-add". The question is, "What two-term expression should we use?" Ultimately, we may be able to solve this problem using many two-term expressions. But there is one that will probably make this easiest: {{{sec(theta) + 1}}}<br>
The reason this particular expression looks promising is that multiplying the current denominator, {{{sec(theta) - 1}}}, by {{{sec(theta) + 1}}} we would have an (a-b)(a+b) pattern and we would get the difference of squares: {{{sec^2(theta) - 1}}} and this difference of squares, according to the {{{tan^2(x) + 1 = sec^2(x)}}} property, could be replaced by {{{tan^2(theta)}}}. With the tan in the numerator this looks very good. So let's go ahead with this plan.<br>
{{{tan(theta)/(sec(theta)-1) = csc(theta) + cot(theta)}}}
{{{(tan(theta)/(sec(theta)-1))((sec(theta)+1)/(sec(theta)+1)) = csc(theta) + cot(theta)}}}
{{{(tan(theta)*(sec(theta)+1))/(sec^2(theta)-1) = csc(theta) + cot(theta)}}}
Replacing the denominator with {{{tan^2(theta)}}}:
{{{(tan(theta)*(sec(theta)+1))/tan^2(theta) = csc(theta) + cot(theta)}}}
The tan in the numerator cancels one of the two tan's in the denominator:
{{{(sec(theta)+1)/tan(theta) = csc(theta) + cot(theta)}}}
Now we can "un-add" to get the two terms we need:
{{{sec(theta)/tan(theta) + 1/tan(theta) = csc(theta) + cot(theta)}}}
The second term is obviously {{{cot(theta)}}} which is what we want! Now we just have to show that the first term on the left is equal to the first term on the right. It may not be immediately obvious how to do this. Another technique that is often used on these identities is: "When in doubt, change any sec, csc, tan or cot into sin and/or cos:
{{{(1/cos(theta))/(sin(theta)/cos(theta)) + cot(theta) = csc(theta) + cot(theta)}}}
If we multiply the numerator and denominator of the fraction by {{{cos(theta)}}} it will cancel the {{{cos(theta)}}}'s in the fraction, leaving:
{{{1/sin(theta) + cot(theta) = csc(theta) + cot(theta)}}}
And since the first term is csc, we are done!
{{{csc(theta) + cot(theta) = csc(theta) + cot(theta)}}}```