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.

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.♦

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.

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.

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.♦

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.♦