Proving Trigonometric Identities

For instance, is an identity.

How do you prove an identity?

To prove an identity, you have to use logical steps to show that one side of the equation can be transformed into the other side of the equation. You do not plug values into the identity to prove anything. There are infinitely-many values you can plug in. Are you really going to prove anything by listing three or four values where the two sides of the equation are equal? Of course not.

And sometimes you'll be given an equation which is not an identity. If you plug a value in where the two sides happen to be equal, such as π/4 for the (false) identity sin(x) = cos(x), you could fool yourself into thinking that a mere equation is an identity. You'll have shot yourself in the foot. So let's not do that.

To prove an identity, your instructor may have told you that you cannot work on both sides of the equation at the same time. This is correct. You can work on both sides together for a regular equation, because you're trying to find where the equation is true. When you are working with an identity, if you work on both sides and work down to where the sides are equal, you will only have shown that, if the starting equation is true, then you can arrive at another true equation. But you won't have proved, logically, that the original equation was actually true.

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Since you'll be working with two sides of an equation (trying to make the one side match the other side), it might be helpful to introduce some notation, if you haven't seen it before. The left-hand side of an equation is denoted by lhs, and the right-hand side is denoted as rhs.

• Prove the following identity:
  
  

In this proof, I did not assume that the equation was an identity; that is, I did not assume that it was true, either for one value of x or for all of them. Instead, I started with one side of the equation and, working only with that side, showed how to convert it to be in the same form as the other side. That final string of equations is what they're wanting for your answer. It is the proof they've asked for.

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"What is the difference", you ask, "between solving and proving? Why can't we work on both sides of the equation at the same time when we're trying to do a proof?" Because of the logic that allows for (or forbids) working with both sides.

When we're working both sides of an equation that we're trying to solve, we are making the assumption that the original equation was true, for at least some value(s) of the variable(s). If we start with a true statement (the equation) and make only logically sound changes to that equation (working on both sides of the equation in exactly the same way), then the result (the solution) must also be true.

However, here we are trying to prove that an equation is true. So we can't use techniques that assume that the equation is true. We instead have to restrict ourselves to working methodically with one side only, and hoping we can figure out a way to transform that side to match the other side. It's a fine distinction, but it does matter.

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• Prove the identity cot(x) + tan(x) = sec(x)csc(x)