## Basic Trigonometric Identities

The basic trigonometric identities come in several varieties. These include the reciprocal identities, ratio identities, Pythagorean identities, symmetric identities, and cofunction identities. Each of these identities follows directly from the definition.

### Reciprocal Identities Theorem

The following identities are true for all values for which they are defined:

 $\sin t=\dfrac{1}{\csc t}$ $\tan t=\dfrac{1}{\cot t}$ $\sec t=\dfrac{1}{\cos t}$ $\cos t=\dfrac{1}{\sec t}$ $\cot t=\dfrac{1}{\tan t}$ $\csc t=\dfrac{1}{\sin t}$

Proof: From the Unit Circle Definition, we have   $\csc t=\dfrac{1}{y}=\dfrac{1}{\sin t}$.   The proofs of the other five identities are similar.♦

### Ratio Identities Theorem

The following identities are true for all values for which they are defined:

 $\sin t=\dfrac{\cos t}{\cot t}$ $\tan t=\dfrac{\sin t}{\cos t}$ $\sec t=\dfrac{\tan t}{\sin t}$ $\sin t=\dfrac{\tan t}{\sec t}$ $\tan t=\dfrac{\sec t}{\csc t}$ $\sec t=\dfrac{\csc t}{\cot t}$ $\cos t=\dfrac{\sin t}{\tan t}$ $\cot t=\dfrac{\cos t}{\sin t}$ $\csc t=\dfrac{\cot t}{\cos t}$ $\cos t=\dfrac{\cot t}{\csc t}$ $\cot t=\dfrac{\csc t}{\sec t}$ $\csc t=\dfrac{\sec t}{\tan t}$

Proof: From the Unit Circle Definition, we have   $\tan t=\dfrac{y}{x}=\dfrac{\sin t}{\cos t}$.   The proofs of the other eleven identities are similar.♦

Of the twelve ratio identities given, only two are commonly cited, namely the ones which involve ratios of sine and cosine.  Besides these twelve ratios, other ratios can be produced and simplified, but either they will either equal the constant 1, or be a product of trig functions.

### Pythagorean Identities Theorem

The following identities are true for all values for which they are defined:

 $\sin^2 t+\cos^2 t=1$ $1+\tan^2 t=\sec^2 t$ $1+\cot^2 t=\csc^2 t$

Proof: The equation of the unit circle is   $x^2+y^2=1$.   Substituting using the Unit Circle Definitions, we obtain the first of the three identities. The other two identities can be obtained by dividing each side of the equation by an appropriate factor.♦

Note that the Pythagorean identities provide a way to express the square of every trigonometric function in an alternate form.

### Symmetric Identities Theorem

The following identities are true for all values for which they are defined:

 $\sin (-t)=-\sin t$ $\tan (-t)=-\tan t$ $\sec (-t)=+\sec t$ $\cos (-t)=+\cos t$ $\cot (-t)=-\cot t$ $\csc (-t)=-\csc t$

Proof: Let the point $A$ on the unit circle have coordinates   $(x,y)$   and arc length $t$, measured counterclockwise from the point   $(1,0)$.   Define point $B$ to be the point on the unit circle whose arc length is $t$, measured clockwise from the point   $(1,0)$.   In other words, measured counterclockwise, the arc length is   $-t$.   Then, by symmetry across the $x$-axis, the coordinates of point $B$ are   $(x,-y)$.   Therefore, we have   $\sin(-t)=-y=-\sin t$.

The proof of the cosine identity is similar. For the tangent identity, we have   $\tan (-t)=\dfrac{\sin(-t)}{\cos(-t)}=\dfrac{-\sin t}{+\cos t}=-\tan t$.

The proofs of the last three trigonometric identities are similar to the proof of the tangent identity.♦

### Cofunction Theorem

The following identities are true for all values for which they are defined:

 $\sin t=\cos\left(\dfrac{\pi}{2}-t\right)$ $\cos t=\sin\left(\dfrac{\pi}{2}-t\right)$ $\tan t=\cot\left(\dfrac{\pi}{2}-t\right)$ $\cot t=\tan\left(\dfrac{\pi}{2}-t\right)$ $\sec t=\csc\left(\dfrac{\pi}{2}-t\right)$ $\csc t=\sec\left(\dfrac{\pi}{2}-t\right)$

Proof: Let the point $A$ on the unit circle have coordinates   $(x,y)$   and arc length $t$, measured counterclockwise from the point   $(1,0)$.   Define point $B$ to be the point on the unit circle whose arc length is   $\dfrac{\pi}{2}-t$,   measured counterclockwise from the same point. Then the coordinates of point $B$ are   $(y,x)$.   Therefore, we have   $\cos\left(\dfrac{\pi}{2}-t\right)=y=\sin t$.

The proofs of the other five identities are similar.♦