# Section 1.4

:::{prf:definition} Identities
:label: idDef

Identities are equations that are true for all values of the variables for which all expression are defined.
:::

:::{prf:definition} Reciprocal Identities
:label: recipID

For all angles $\theta$ for which both functions are defined, the following hold true:

\begin{align*}
    \sin (\theta) & = \frac{1}{\csc(\theta)} & \cos(\theta) & = \frac{1}{\sec(\theta)} & \tan(\theta) & = \frac{1}{\cot(\theta)}\\
    \csc(\theta) & = \frac{1}{\sin(\theta)} & \sec(\theta) & = \frac{1}{\cos(\theta)} & \cot(\theta) & = \frac{1}{\tan(\theta)}
\end{align*}
:::

Remember in general $f(5x)\ne 5f(x)$. This is also true for trig functions. 

$$\cos(2x) \ne 2\cos(x)$$

Similarly,

$$\cos(x+y) \ne \cos(x)+\sin(y)$$

The signs for the sine and cosine function are as follows.

!['image of the sign for sine and cosine'](images/signTrigValues.png)

Let $k$ be any integer.

|Trig Function | Domain | Range |
|---|:-:|:-:|
|$\sin(\theta)$,$\cos(\theta)$| $\mathbb{R}$|$[-1,1]$|
|$\tan(\theta)$| $x\ne \frac{\pi}{2}+\pi k$|$\mathbb{R}$|
|$\cot(\theta)$| $x\ne \pi + \pi k$|$\mathbb{R}$|
|$\sec(\theta)$| $x\ne \frac{\pi}{2} + \pi k$|$(-\infty,-1]\cup[1,\infty)$|
|$\csc(\theta)|$x\ne \pi + \pi k$|$(-\infty,-1]\cup[1,\infty)$|


:::{prf:theorem} Quotient Identities
:label: QuotId

For all angles $\theta$ for which the denominators are not zero, the follow are true.

\begin{align*}
    \frac{\sin(\theta)}{\cos(\theta)} & = \tan(\theta) & \frac{\cos(\theta)}{\sin(\theta)} & = \cot(\theta)
\end{align*}
:::

:::{prf:theorem} Pythagorean Identities
:label: trigPythThm

For all angles $\theta$ for which the function values are defined, the following are true.

$$\sin^2(\theta)+\cos^2(\theta) = 1$$

$$\tan^2(\theta)+1=\sec^2(\theta)$$

$$\cot^2(\theta)+1=\csc^2(\theta)$$

:::