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Reciprocal Identities

Written by tutor Jeffery D.

A brief summary

The reciprocal of a term a/b is defined as b/a.

If we let a = sin(x) and b = 1, and then take the reciprocal we obtain 1/sin(x).

Similarly, if we let a = cos(x) and b = 1, and then take the reciprocal we obtain 1/cos(x).

Recall that tan(x) is defined as sin(x)/cos(x). Taking the reciprocal we obtain cos(x)/sin(x).

All of the reciprocals of the trigonometric functions can be proven geometrically using various approaches (look into this with an image search). These derivations of the reciprocal trigonometric functions on the unit circle (including the derivation of the tangent function using properties of similar triangle) are where we get their names (in case you were wondering). For example, the reciprocal of the cosine function is called the secant function. In one derivation, the secant line is drawn from the origin of the xy – plane and it cuts through the unit circle to become the hypotenuse of a triangle formed with the line x = 1, which is vertically tangent to the unit circle (the tangent) as one of its sides. Secant means, “to cut”. Using the properties of similar triangles it can be shown that the ratio of the hypotenuse (which has a length of 1) and the cosine (which is the base) is equivalent to the ratio of the (secant) line from the origin intersecting with the tangent (the secant line), and its “base”, which is 1. This is just one approach and is interesting to those who are mathematically inclined. There are many figures and pictures that can illustrate this more elegantly.

But, what a person in a trigonometry class needs to know are the following:

The Reciprocal Identities:

1/sin(x) = csc(x) (where csc(x) is the cosecant function).

1/cos(x) = sec(x) (where sec(x) is the secant function).

1/tan(x) = cot(x) (where cot(x) is the cotangent function).

The reciprocal identities can be derived from the Pythagorean identity. The Pythagorean identity tells us

sin2(t) + cos2(t) = 1.         (1)

If we divide the entire equation (1) first by sin2(t), we can get the resulting equation,

1 + cot2(t) = csc2(t)

If we divide the entire equation (1) by cos2(t), we obtain the following equation,

tan2(t) + 1 = sec2(t)

We use these identities to make difficult things easier by substituting equivalent expressions. You will be asked to do things like: simplify using identities. Later, in calculus, you can use these in to make integrations easier.

Example:
Use identities to simplify:

cos(x)/tan(x)

Substitute sin(x)/cos(x) for tan(x)

and we obtain: cos(x)/sin(x)/cos(x)

Now we can do a little more work: cos(x) * cos(x)/sin(x)

Recall that cos(x)/sin(x) = cot(x)

So, the simplest we can make this is the form: cos(x) cot(x).

A word of warning: Do not confuse the notation for the inverse trigonometric functions with the reciprocal functions.

Example: sin(x)-11/sin(x)

*sin(x)-1 = arcsin. A portion of the sine function inverted with a restricted domain.

Reciprocal Identities Practice Quiz

tan(x)/sin(x) = sec(x)

A. True
B. False
The correct answer here would be A.

sin2(x)/tan(x) = sin(x)cos(x)

A. True
B. False
The correct answer here would be A.
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