Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 noun The reciprocal of the sine of an angle in a right triangle.
 noun The secant of the complement of a directed angle or arc.
from The Century Dictionary.
 noun In trigonom., the secant of an angle or arc equal to the difference between a given angle or are (whose cosecant it is) and 90°; the secant of the complement of the given angle or arc. See
complement .
from the GNU version of the Collaborative International Dictionary of English.
 noun (Trig.) The secant of the complement of an arc or angle. See
Illust. offunctions .
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun trigonometry In a
right triangle , thereciprocal of thesine of anangle . Symbols:cosec ,csc
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 noun ratio of the hypotenuse to the opposite side of a rightangled triangle
Etymologies
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Examples

Add together the secant of the Latitude, the cosecant of the P.D., the cosine of the Half Sum and the sine of the Difference (Table 44).

After the creation of the universe, light speed declined following a curve approximating the curve of the cosecant squared.

On this topic, it isn't completely clear to me why we make kids memorize what the secant, cosecant, and cotangent are.

If your calculator does not have keys for the cosecant (csc), secant (sec), or cotangent (cot) functions, ﬁ rst

Graph of the secant function for values of x between 3 rad and 3 rad. is the same as the range of the cosecant function.

They are known as the hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, and hyperbolic cotangent.

If we are operating on some variable x, the arctangent of x is denoted tanÀ1 (x) or arctan (x) The inverse of the cosecant function is the arccosecant function.

This should not come as a surprise, because the cosecant and secant functions are the reciprocals of the sine and cosine functions, respectively, and the sine and cosine are horizontally displaced by 1 = 4 cycle.

SOLUTION 41 Remember that the hyperbolic cosecant (csch) is the reciprocal of the hyper  bolic sine (sinh).

The cosecant and secant functions have the same general shape, but they are shifted by 90 (/2 rad), or 1 = 4 cycle, with respect to each other.
chained_bear commented on the word cosecant
"...the boys were taught, with words and diagrams, the nature of sine, cosine, tangent, cotangent, secant and cosecant, the relations between them, and their value in helping to find your position in a prodigious ocean, no shore, no landmark for ten thousand miles."
P. O'Brian, The Yellow Admiral, 145
March 19, 2008