When simplifying problems that have reciprocal trig functions, start by substituting in the identities for each. If possible, write tangent in terms of sine and cosine. Use algebra to eliminate any complex fractions, factor, or cancel common terms. When using trigonometric identities, make one side of the equation look like the other or work on both sides of the equation to arrive at an identity (like 1=1).
Trigonometric identities are really important as you've probably already seen if you've used them to graph. One of the things that you'll find in your homework occasionally is problems that ask you to simplify a trig expression. Well let's recall some basic trig identities that will be need right now. These are some of the reciprocal identities. Secant theta equals 1 over cosine theta, cosecant theta is 1 over sine theta and tangent theta is sine theta over cosine theta. So let's use that to simplify secant theta plus 1 over sine theta plus tangent theta. I'll write that over here. Secant theta plus 1, sine theta plus tangent theta. A lot of times when you're dealing with um some of the reciprocal trig functions, the best strategy is to switch to sines and cosines. So let me do that first. I'll replace secant theta with 1 over cosine theta. And I'll replace tangent theta with sine theta over cosine theta. Now, at first glance, this is actually a little more complicated looking than we started but now that we've got sines and cosines we can simplify this using Algebra. Notice that I have a complex fraction here and the denominators in the little fractions are cosine theta so I can get rid of them by multiplying the top and bottom by cosine theta. Now remember when I do this, this cosine theta is going to be multiplied across both of these terms and this cosine theta will be multiplied across both of these terms. So in the numerator, I get cosine times 1 over cosine is 1. Cosine times 1 is cosine. Cosine times sine is sine cosine and sine over cosine times cosine the cosine's cancel and you get sine. And this is really good because notice, I can factor a sine theta out of the bottom and I'm left with cosine theta plus 1 which is exactly what I have in the numerator. And so these things cancel. And I get 1 over sine theta. Now when you're simplifying a trigonometric expression or any expression, it's best to have as few operations and as few functions in your final answer as possible. I still have the operation of division and I can get rid of that if I just replace this with cosecant theta. So this whole big expression reduces down to the simple cosecant theta. So that's what we mean by simplifying a trigonometric expression. Don't forget. If you're dealing with reciprocal trig functions like secant tangent, sometimes it's best to switch to sine and cosine. And don't forget this trick, when you have complex fractions, you can do what we call fraction busting. Multiply by the common denominator of the little fractions.