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The tangent function, denoted as \(\tan \theta\), is a fundamental function in trigonometry. It can be defined in terms of sine and cosine functions as:
\( \tan \theta = \frac{ \sin \theta }{ \cos \theta } \).
This means the tangent of an angle is the ratio of the sine of the angle to the cosine of the angle.

The tangent function has a few key properties:

• It is periodic with a period of \( \pi \). This means it repeats its values every \( \pi \) units.
• It has vertical asymptotes where \( \cos \theta = 0 \), which are at \( \theta = \frac{\pi}{2} + k\pi\) for any integer \( k \).

In the interval \(0 \leq \theta < 2 \pi\), the tangent function is equal to 1 at two points:

• \( \theta = \frac{\pi}{4} \)
• \( \theta = \frac{5\pi}{4} \)

These are critical values needed for solving our trigonometric equation.

The secant function, denoted as \( \sec \theta \), is another important trigonometric function. It is defined as the reciprocal of the cosine function:
\( \sec \theta = \frac{1}{\cos \theta} \).

Here are some essential properties of the secant function:

• It is undefined where \( \cos \theta = 0 \), which are at \( \theta = \frac{\pi}{2} + k\pi \) for any integer \( k \).
• It is periodic with a period of \( 2\pi \).

In the context of our equation, we specifically need to know where the secant function equals 1. This occurs when the cosine function equals 1:

• In the interval \(0 \leq \theta < 2\pi\), \( \cos \theta = 1 \) at \( \theta = 0 \).

This helps us solve our equation by identifying another valid value of \( \theta \).

Solving trigonometric equations involves finding all angles that satisfy a given equation within a specific interval.
The steps to solve trigonometric equations generally include:

• Understanding and simplifying the given equation.
• Identifying specific trigonometric values for the given functions.
• Finding all solutions within the given interval, taking into account the periodic nature of trigonometric functions.

In our example, we are solving the equation \((\tan \theta - 1)(\sec \theta - 1) = 0\).
This is solved by breaking it down into two separate equations: \( \tan \theta - 1 = 0 \) and \( \sec \theta - 1 = 0 \).
We determine the solutions for each function individually and then combine them:

• \( \tan \theta = 1\) yields \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \).
• \( \sec \theta = 1 \) yields \( \theta = 0 \).

Therefore, the combined solutions within the interval \(0 \leq \theta < 2\pi\) are \( \theta = 0 \), \( \theta = \frac{\pi}{4} \), and \( \theta = \frac{5\pi}{4} \).
Mastering the approach to solving such equations can significantly improve your mathematical skills.