Note that more detail about **solving trig equations** can be found here in the **Solving Trigonometric Equations** section.

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

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 $ \displaystyle \csc (x)=\frac{1}{\sin (x)}$; for **any value** of $ x$, this equation is true.

**Trigonometric 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.

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.

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.

And note that there may be more than one way to do these! Several answers may be “right”. And, of course, $ x$ and $ \theta$, as well as other variables are exchangeable.

## 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$:

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

- Match trig functions; for example, if you have a $ \cos \text{, }{{\cos }^{2}}$, and $ {{\sin }^{2}}$, turn the $ {{\sin }^{2}}$ into $ \left( {1-{{{\cos }}^{2}}} \right)$.
- Use
**common****denominators**to**combine terms**. - 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. - (To do this, multiply by
**1**, or $ \displaystyle \frac{{\text{conjugate}}}{{\text{conjugate}}}$, as shown below.) - Turn all trig functions into
**sin**and**cos**if you have other trig functions, such as**tan.**

Here are a few more that are more complicated:

## 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 $ \left[ {0,2\pi } \right)$:

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

## Sum and Difference Identities

The **sum and difference identities** are used to split up angles to find easier values (for example, on the unit circle). The identities are also used 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.

First, simplify and find exact values using **the sum and difference identities**:

Here are some **sum and difference identity** proofs. The last two are quite tricky!

## 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:

## Double Angle and Half Angle Identities

The **double angle and half angle identities** are used to split up angles to make easier values (for example, on the unit circle). The identities are also used 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:

Here are some **double and half angle identity** proofs. Notice how we always try to start on the **more complicated side**.

Here’s one more that’s difficult; sometimes it’s easier to start down both sides to see how to prove the identity:

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:

## 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:

## 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:

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. For example, if you have a $ {{\tan }^{2}}$ on one side, and a $ \displaystyle \frac{{\sin }}{{\cos }}$ on the other, change the $ \displaystyle \frac{{\sin }}{{\cos }}$ to a**tan**. - 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-hand 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 doesn’t match on each side. For example, if you have two terms on the left-hand side and only one term on the right-hand side, find the common denominator and add the two terms on the left-hand 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, $ \displaystyle \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 \left( {2A} \right)$, to know which version of identity to use, check to see if there’s a “$ -1$” or a “$ +1$” on same side of the identity; you’ll probably want to cancel these out. For example, use $ 1-2{{\sin }^{2}}x$ if there’s a “$ -1$” following the $ \cos \left( {2A} \right)$ (then you’ll end up with $ 1-2{{\sin }^{2}}x-1=-2{{\sin }^{2}}x$), and use $ 2{{\cos }^{2}}x-1$ if there’s a “$ +1$” following the $ \cos \left( {2A} \right)$ (then you’ll end up with $ 2{{\cos }^{2}}x-1+1=2{{\cos }^{2}}x$). If there’s a “$ -\,\,{{\cos }^{2}}x$”, or a “$ +\,\,{{\sin }^{2}}x$” following the $ \cos \left( {2A} \right)$, 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$. - With
**quadratics**, get everything to one side and try to factor. - 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 (
**GCF**s) if can. - When
**solving**, simplify with identities first, if you can. - When
**solving**, you can square each side, but don’t divide both sides by factors with variables, since you might be missing out on solutions. If you need to cross-multiply, or multiply both sides by what’s in a denominator (even when one side equals**0**), make sure you’re not missing solutions. (It might be a good idea to see how many solutions there are in a graphing calculator if you can). And always check for**extraneous solutions**:**solutions must work in the original equations**, and**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).

Here are some **Mixed Identity Proof** problems:

Here are more **Mixed Identity Proofs**:

**Understand these problems, and practice, practice, practice!**

**For Practice**: Use the **Mathway** widget below to try a **Trig Identity** problem. Click on **Submit** (the blue arrow to the right of the problem) and click on **Verify the Identity **to see the answer**.**

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