Basic Trigonometric Identities

The basic trigonometric identities are equations involving the trigonometric functions (sine, cosine, tangent, secant, cosecant, and cotangent) that are always true regardless of the variables involved. We can use these equations to manipulate expressions involving trig and make them easier to work with.

In this article, we'll look at trigonometric functions derived from the Pythagorean Theorem, the reciprocal identities, and the quotient identities. Let's get started!

Basic trigonometric identities: Pythagorean Theorem

This category includes three of the most commonly-used trigonometric identities:

1. ${\sin }^2\left(x\right)+{\cos }^2\left(x\right)=1$

2. $1+{\tan }^2\left(x\right)={\sec }^2\left(x\right)$

3. $1+{\cot }^2\left(x\right)={\csc }^2\left(x\right)$

We can use these to simplify trigonometric identities. For instance:

$1-\frac{{\sin }^2\left(\theta \right)}{{\tan }^2\left(\theta \right)}$

We can start by rewriting the tangent as sin/cos:

$1-\frac{{\sin }^2\left(\theta \right)}{\left[\frac{{\sin }^2\left(\theta \right)}{{\cos }^2\left(\theta \right)}\right]}$

Simplify:

$1-{\sin }^2\left(\theta \right)\frac{{\cos }^2\left(\theta \right)}{{\sin }^2\left(\theta \right)}$

$1-{\cos }^2\left(\theta \right)$

Finally, we can apply the fundamental Pythagorean identity:

${\sin }^2\left(\theta \right)$

The Pythagorean Theorem is always one of the stronger tools in the mathematician's toolbox, and the example above illustrates what we can do with it nicely.

Basic trigonometric identities: reciprocal identities

The reciprocal identities give us equivalent equations for each of the six trigonometric functions:

1. $\sin \left(x\right)=\frac{1}{\csc \left(x\right)}$

2. $\csc \left(x\right)=\frac{1}{\sin \left(x\right)}$

3. $\cos \left(x\right)=\frac{1}{\sec \left(x\right)}$

4. $\sec \left(x\right)=\frac{1}{\cos \left(x\right)}$

5. $\tan \left(x\right)=\frac{1}{\cot \left(x\right)}$

6. $\cot \left(x\right)=\frac{1}{\tan \left(x\right)}$

Let's take a closer look at how these work by proving the following is true:

${\sec }^2\left(\theta \right)+{\csc }^2\left(\theta \right)={\sec }^2\left(\theta \right){\csc }^2\left(\theta \right)$

First, we can use the reciprocal identities above to rewrite the right side of the expression:

${\sec }^2\left(\theta \right){\csc }^2\left(\theta \right)=\frac{1}{{\cos }^2\left(\theta \right)}\times \frac{1}{{\sin }^2\left(\theta \right)}$

Multiply these two fractions:

${\sec }^2\left(\theta \right)+{\csc }^2\left(\theta \right)=\frac{1}{{\cos }^2\left(\theta \right)\times {\sin }^2\left(\theta \right)}$

Apply the reciprocal to the left hand sides:

$\frac{1}{{\cos }^2\left(\theta \right)}+\frac{1}{{\sin }^2\left(\theta \right)}=\frac{1}{{\cos }^2\left(\theta \right)\times {\sin }^2\left(\theta \right)}$

Put the left hand side over a common denominator:

$\frac{{\cos }^2\left(\theta \right)+{\sin }^2\left(\theta \right)}{{\cos }^2\left(\theta \right)\times {\sin }^2\left(\theta \right)}=\frac{1}{{\cos }^2\left(\theta \right)\times {\sin }^2\left(\theta \right)}$

Now, apply the Pythagorean identity ${\sin }^2\left(x\right)+{\cos }^2\left(x\right)=1$ :

$\frac{1}{{\cos }^2\left(\theta \right)\times {\sin }^2\left(\theta \right)}=\frac{1}{{\cos }^2\left(\theta \right)\times {\sin }^2\left(\theta \right)}$

Now that the left-hand side and right-hand side match, we have proven the identity.

Basic trigonometric identities: quotient identities

The quotient identities allow us to rewrite expressions involving tan and cot in terms of sin and cos, making them easier to work with:

$\tan \left(u\right)=\frac{\sin \left(u\right)}{\cos \left(u\right)}$

$\cot \left(u\right)=\frac{\cos \left(u\right)}{\sin \left(u\right)}$

You probably knew these even before you knew to call them "quotient identities," and we even rewrote a tangent as sine/cosine in the first example above. That said, their applications can get fairly complex. Let's verify the identity of $\cos \left(\theta \right)+\sin \left(\theta \right)\tan \left(\theta \right)=\sec \left(\theta \right)$ :

Consider the expression on the left side of the following equation:

$\cos \left(\theta \right)+\sin \left(\theta \right)\tan \left(\theta \right)=\cos \left(\theta \right)\frac{\cos \left(\theta \right)}{\cos \left(\theta \right)}+\sin \left(\theta \right)\frac{\sin \left(\theta \right)}{\cos \left(\theta \right)}$

Using the quotient identity, we can turn that into:

${\cos }^2\left(\theta \right)+\frac{{\sin }^2\left(\theta \right)}{\cos \left(\theta \right)}$

Whenever we see cosines or sines squared, there's probably an application of a Pythagorean Identity waiting to be found:

$\frac{1}{\cos \left(\theta \right)}$

Finally, we can apply a reciprocal identity:

$\sec \left(\theta \right)$

We started with a lot of terms, but we got it down to a single trigonometric function by the end. That is what trigonometric identities are all about.

Practice problems

a. Express without using any fractions: $\frac{1}{\cot \left(x\right)}+\frac{1}{\tan \left(x\right)}+\frac{1}{\sin \left(x\right)}$

This is an application of the reciprocal identities. The $\frac{1}{\cot \left(x\right)}$ term can be simplified to $\tan \left(x\right)$ , the $\frac{1}{\tan \left(x\right)}$ to $\cot \left(x\right)$ , and the $\csc \left(x\right)$ to . Using those, we can rewrite the expression like this:

$\tan \left(x\right)+\cot \left(x\right)+\csc \left(x\right)$

b. Rewrite $\csc \left(200\right)$ in terms of sin and cos.

We can apply the quotient identity to this problem, rewriting $\csc \left(u\right)$ as $\frac{\sin \left(u\right)}{\cos \left(u\right)}$ $\frac{\sin \left(u\right)}{\cos \left(u\right)}$ . Since $u=200$ , we get:

$\tan \left(200\right)=\frac{\sin \left(200\right)}{\cos \left(200\right)}$

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