As a mathematical expert, I'm delighted to delve into the fascinating world of trigonometric identities. Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right-angled triangles. It's a fundamental part of geometry and has wide applications in various fields such as physics, engineering, and computer science.

One of the key concepts in trigonometry is the relationship between the trigonometric functions of complementary angles. The term "complementary" refers to two angles whose measures add up to 90 degrees. For instance, if you have an angle \( \theta \), its complement is \( 90^\circ - \theta \) or \( \frac{\pi}{2} - \theta \) in radians.

The trigonometric functions sine (\( \sin \)), cosine (\( \cos \)), tangent (\( \tan \)), cotangent (\( \cot \)), secant (\( \sec \)), and cosecant (\( \csc \)) can all be expressed in terms of their complementary angles. This is particularly useful when dealing with problems where you might have an angle and its complement, as it allows you to use the known values of one function to find the value of another.

Now, let's focus on the tangent function. The tangent of an angle \( \theta \) is defined as the ratio of the sine of \( \theta \) to the cosine of \( \theta \), which can be written as:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]

However, when we consider the complementary angle \( 90^\circ - \theta \), we can express the tangent of \( \theta \) in terms of the sine and cosine of its complement. The identity that relates the tangent of an angle to the sine and cosine of its complement is:
\[ \tan(\theta) = \cot(90^\circ - \theta) \]

This identity is derived from the fact that the tangent of an angle is the reciprocal of the cotangent of its complement. The cotangent function is defined as the reciprocal of the tangent, so:
\[ \cot(\theta) = \frac{1}{\tan(\theta)} \]

Substituting the definition of tangent into the cotangent, we get:
\[ \cot(\theta) = \frac{1}{\frac{\sin(\theta)}{\cos(\theta)}} = \frac{\cos(\theta)}{\sin(\theta)} \]

Now, if we take \( \theta \) and replace it with its complement \( 90^\circ - \theta \), we get:
\[ \cot(90^\circ - \theta) = \frac{\cos(90^\circ - \theta)}{\sin(90^\circ - \theta)} \]

Using the co-function identities, which state that:
\[ \cos(90^\circ - \theta) = \sin(\theta) \]
\[ \sin(90^\circ - \theta) = \cos(\theta) \]

We can rewrite the cotangent of the complement as:
\[ \cot(90^\circ - \theta) = \frac{\sin(\theta)}{\cos(\theta)} \]

Which is the definition of the tangent of \( \theta \). Thus, we have established that:
\[ \tan(\theta) = \cot(90^\circ - \theta) \]

This relationship is particularly useful in trigonometry because it allows us to switch between the tangent and cotangent functions by considering the complementary angles. It's also a good example of how trigonometric identities can simplify calculations and provide alternative methods for solving problems.

The periodicity of trigonometric functions is another important aspect to consider. Sine and cosine functions have a period of \( 2\pi \) radians (or \( 360^\circ \)), which means they repeat their values every \( 2\pi \) radians. Secant and cosecant also have a period of \( 2\pi \), but they are undefined at the points where their denominators (cosine and sine, respectively) are zero. Tangent and cotangent, on the other hand, have a period of \( \pi \) radians (or \( 180^\circ \)), as they repeat their values every \( \pi \) radians. This is because the tangent function has asymptotes where it is undefined, specifically at \( \frac{\pi}{2} + k\pi \) for any integer \( k \), and the cotangent function has similar behavior.

In summary, the tangent function can be expressed in terms of its complementary angle using the identity \( \tan(\theta) = \cot(90^\circ - \theta) \). This identity, along with the understanding of periodicity and the behavior of trigonometric functions, is crucial for solving a wide range of problems in trigonometry.

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Identities expressing trig functions in terms of their complements. ... Each of the six trig functions is equal to its co-function evaluated at the complementary angle. Periodicity of trig functions. Sine, cosine, secant, and cosecant have period 2-- while tangent and cotangent have period --.read more >>