You are given a list of Trig Identities. One of those identities is

### Trigonometric Identities

**Trigonometric identities** are true for any value of

**Reciprocal Identities:**

Other identities involve the tangent, variations on the Pythagorean Theorem, phase shifts, and negative angles. We will discover them in this concept.

We know that **Tangent Identity**.

Whenever we are trying to verify, or prove, an identity, we start with the statement we are trying to prove and work towards the desired answer. In this case, we will start with

Then, rewrite the complex fraction as a division problem and simplify.

We now have what we wanted to prove and we are done. **Once you verify an identity, you may use it to verify other identities.**

Now, let's show that

Change the sine and cosine in the equation into the ratios. In this problem, we will use

Now,

This is one of the **Pythagorean Identities** and very useful.

Finally, let's verify that

The function

The red function above is **Cofunction Identity**.

### Examples

#### Example 1

Earlier, you were asked to prove the identity of

First, recall that

Now, if we have

We know that

We can now flip this identity around to get:

#### Example 2

Prove the Pythagorean Identity:

First, let’s use the Tangent Identity and the Reciprocal Identity to change tangent and secant in terms of sine and cosine.

Now, change the 1 into a fraction with a base of

In the second to last step, we arrived at the original Pythagorean Identity

#### Example 3

Without graphing, show that

First, recall that

Now, if we have

### Review

- Show that
cotθ=cosθsinθ . - Show that
tanθ=secθcscθ . - Show that
1+cot2θ=csc2θ . - Explain why
cos(π2−θ)=sinθ by using the graphs of the two functions. - Show that
sec(−θ)=secθ . - Explain why \begin{align*}\tan (- \theta)=- \tan \theta\end{align*} is true, using the Tangent Identity and the other Negative Angle Identities.

Verify the following identities.

- \begin{align*}\cot \theta \sec \theta=\csc \theta\end{align*}
- \begin{align*}\frac{\cos \theta}{\cot \theta}=\frac{\tan \theta}{\sec \theta}\end{align*}
- \begin{align*}\sin \theta \csc \theta=1\end{align*}
- \begin{align*}\cot(- \theta)=- \cot \theta\end{align*}
- \begin{align*}\tan x \csc x \cos x=1\end{align*}
- \begin{align*}\frac{\sin^2 \left(-x\right)}{\tan^2 x}= \cos ^2 x\end{align*}

Show that \begin{align*}\sin^2 \theta+ \cos^2 \theta=1\end{align*} is true for the following values of \begin{align*}\theta\end{align*}.

- \begin{align*}\frac{\pi}{4}\end{align*}
- \begin{align*}\frac{2 \pi}{3}\end{align*}
- \begin{align*}- \frac{7 \pi}{6}\end{align*}
- Recall that a function is odd if \begin{align*}f(-x)=-f(x)\end{align*} and even if \begin{align*}f(-x)=f(x)\end{align*}. Which of the six trigonometric functions are odd? Which are even?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 14.6.