Trigonometric Identities

This section covers:

Before we get started, here is a table of some common trig identities for future reference:

Summary of Trig Identities

Mirror Picture

An “identity” is something that is always true, so you are typically either substituting or trying to get two sides of an equation to equal each other.   Think of it as a reflection; like looking in a mirror.  An example of a trig identity is  \(\csc (x)=\frac{1}{\sin (x)}\); for any value of x, this equation is true.

Trig identities are sort of like puzzles since you have to “play” with them to get what you want.  You will also have to do some memorizing for these, since most of them aren’t really obvious.  You may not like Trig Identity problems, since they can resemble the proofs that you had to in Geometry.  I actually love them, since I love to do puzzles!

There are typically two types of problems you’ll have with trig identities: working on one side of an equation to “prove” it equals the other side, and also solving trig problems by substituting identities to make the problem solvable.

We’ll start out with the simpler identities that you’ve seen before.

Reciprocal and Quotient Identities

You’ve already seen the reciprocal and quotient identities.  You can also write these as “sin x”, and so on.

Reciprocal Quotient Identities

Here are some examples of simple identity proofs with reciprocal and quotient identities.  Typically, to do these proofs, you must always start with one side (either side, but usually take the more complicated side) and manipulate the side until you end up with the other side.  (Some teachers will let you go down both sides until the two sides are equal).

The best way to solve these is to turn everything into sin and cos.  Note how we work on one side only and pull down the other side when it matches.  It doesn’t matter which side we start on, but typically, it’s the most complicated.

Also note that sometimes we have to find common denominators, like in the last example.  We didn’t need to turn it into sin and cos, since we only had tan and cot in the identity (although it still would have worked).

Reciprocal Quotient Identity Examples

Pythagorean Identities

The Pythagorean identities are derived from (you guessed it!) the Pythagorean Theorem.  Going back to the unit circle, notice that   \({{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\):

Pythagorean Identifies Proof

Here are some examples of solving Pythagorean Identities.  To “prove” the identities, we use the following “tricks” if we can:

  1. Match trig functions; for example, if you have a  \(\cos \text{, }{{\cos }^{2}}\text{, and }{{\sin }^{2}}\), turn the  \({{\sin }^{2}}\)  into  \(\left( {1-{{{\cos }}^{2}}} \right)\).
  2. Use common denominators to combine terms.
  3. Use conjugates (the first term and then change sign and then second term) to multiply numerator or denominator with two terms (binomials). This will create a difference of two squares to work with.
  4. Turn all trig functions into sin and cos if you have other trig functions, such as tan

Pythagorean Identities

And here are a few more that are more complicated:

More Pythagorean Identities

Solving with Reciprocal, Quotient and Pythagorean Identities

Here are some problems where we have use reciprocal and/or Pythagorean identities to solve trig equations in the interval [0, 2π):

Solving Trig Equations with Identities

Here’s another one where we have to check for extraneous solutions.  Solve over the reals:

Solving Trig Extraneous Solutions

Sum and Difference Identities

We use sum and difference identities when we need to split up the angle to make it easier to find the values (for example, to find values on the unit circle).

We also use the identities in conjunction with other identities to prove and solve trig problems.

Here are the sum and difference identities, and tricks to help you memorize them.

Sum and Difference Identities

First, let’s solve some problems using the sum and difference identities:

Sum and Difference Identify Problems

And let’s do some sum and difference identity proofs.  The last two are quite tricky!

Sum and Difference Identity Proofs

Solving with Sum and Difference Identities

Here are some problems where we use Sum and Difference identities to solve trig equations in the indicated interval:

Solving with Sum and Difference Identities

Double Angle and Half Angle Identities

We use double angle and half angle identities the same way we used sum and difference identities when we need to split up the angle to make it easier to find the values (for example, to find values on the unit circle).

We also use the identities in conjunction with other identities to prove and solve trig problems.

Here are the double angle and half angle identities, and tricks to help you memorize them:

Double and Half Angle Identities

Let’s do some double and half angle identity proofs.  Notice how we always try to start on the more complicated side.

Double and Half Angle Identity Proofs

Now let’s use these identities to find exact values for the following expressions using triangles, similar to what we did here in the in The Inverse Trigonometric Functions section:

Double and Half Angle Identity Triangle Problems

Solving with Double and Half Angle Identities

Here are some problems where we have use Double and Half Angle identities to solve trig equations in the indicated interval:

Solving Trig Problems with Double and Half Angle Identities

Trig identity Summary and Mixed Identity Proofs

Now let’s put it all together.  First, here is a table with all the identities we’ve talked about:

Summary of Trig Identities

Here are a set of "hints” that might help you prove and solve trig identity problems:

  • Start with the more complicated side. If you absolutely can’t get to the other side, go down both sides, see where the two sides are identical and then move up one of the sides.  Some teachers will let you work down both sides until the two sides match up.
  • Turn everything into sin and cos, for example if you have tan or reciprocal functions that you can simplify.
  • Match trig functions (like tan) to what’s on the other side.
  • Look at the other side of the identity to see what direction to go on the more complicated side. For example, if there is one term on the right side, strive for one term on the left.
  • Use Pythagorean Identities when you see you can cancel something out (like a "1”) or you see a trig function that is squared that you can eliminate.
  • Find common denominators if the number of terms don’t match on each side. For example, if you have two terms on the left side and only one term on the right side, find the common denominator and add the two terms on the left side so they become one.   If you have two terms on both sides, for example, you may want to leave them alone. You may also need to "break apart” terms when there is more than one term in the numerator (using the same denominator), for example,  \(\frac{{x+2}}{x}=\frac{x}{x}+\frac{2}{x}=1+\frac{2}{x}\). It’s a good idea to simplify fractions (for example, using reciprocal identities) before finding common denominators and adding or subtracting fractions).
  • Divide numerator and denominator by something that makes the term simplified, for example, if you have a difference of squares in the numerator, divide numerator and denominator by one of these factors.
  • Cross out (simplify) anything you can earlier rather than later.
  • For cos(2A), to know which version of identity to use, check to see if there’s a "– 1” or a "+ 1” on same side; you’ll probably want to cancel these out. For example, use  \(1-2{{\sin }^{2}}x\)   if there’s a "– 1” following, and use  \(2{{\cos }^{2}}x-1\)  if there’s a  "+ 1” following.  If there’s a   "\(-\,\,{{\cos }^{2}}x\)”,  or a   "\(+\,\,{{\sin }^{2}}x\)”  following, you may want to use  \({{\cos }^{2}}x-{{\sin }^{2}}x\), to be able to simplify.
  • Watch for difference of squares, such as  \(\left( {\cos x-1} \right)\left( {\cos x+1} \right)={{\cos }^{2}}x-1\).
  • Multiply by conjugates, usually in denominators, but sometimes in numerators, to get difference of squares.  (Multiply by 1, with the conjugate in both the numerator and denominator).
  • Factor out Greatest Common Factors (GCFs) if can.
  • Again, if there’s a random "1” or "–1” that you need to get rid of, find an identity that, when added to it, eliminates it.
  • When solving, simplify with identities first, if you can.
  • When solving, you can square each side, or multiply both sides by what’s in a denominator, but check for extraneous solutions (denominators can’t be 0).
  • When solving, if you get answers for any trig function that has asymptotes (like tan), check for extraneous solutions (solutions that would be asymptotes). Also check for solutions that make a denominator 0; these should be eliminated.

Here are some Mixed Identity Proof problems:

Mixed Identity Proofs

Here are more Mixed Identity Proofs:

More Mixed Identity Proofs

And a couple more:

Even More Mixed Identity Proofs

Understand these problems, and practice, practice, practice!


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On to Law of Sines and Cosines, and Areas of Triangles  – you’re ready! 

2 thoughts on “Trigonometric Identities

    • Here’s an identity that’s a little more difficult – do you want to try it? I hope to put it up on my site soon. (1 + sinx + cosx)/(1 – sinx + cosx)= (1 + sinx)/cosx
      Thanks, Lisa

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