Understanding the Tangent Function: Definition, Properties, and Identities

tanx

sec^2x

The function tan(x) represents tangent of angle x. It is defined as the ratio of the sine of x to the cosine of x:

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

It is important to note that the tangent function is undefined at certain values of x, where the cosine function equals zero. These values are of the form π/2 + πk, where k is an integer. Therefore, we say that the tangent function has vertical asymptotes at these values.

Additionally, the range of tan(x) is all real numbers, except for the values between -π/2 and π/2, where the cosine function is positive and the tangent function is increasing. Outside of this range, the tangent function is decreasing.

There are several useful identities involving the tangent function, such as:

1. tan(x + π) = tan(x)
2. tan(-x) = -tan(x)
3. tan(x) = sin(x) / cos(x) = 1 / cot(x)

These identities can be used to simplify trigonometric expressions involving the tangent function.

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