If \(\tan A=\frac{a}{b}, a>0, b>0,\) and \(0

Trigonometric identities are equations involving trigonometric functions that hold true for all values within the domains of the functions involved. They are essential in simplifying expressions, solving equations, and precisely define the relationships between the trigonometric functions. For example, the Pythagorean identities, which include formulas like \(\sin^2 A + \cos^2 A = 1\),\(\tan^2 A + 1 = \sec^2 A\), and \(1 + \cot^2 A = \csc^2 A\) are all derived from the Pythagorean Theorem applied to a right triangle with a unit hypotenuse.

Understanding how to manipulate these identities can be the key to solving many trigonometric problems. For instance, knowing that \(\tan A = \frac{\sin A}{\cos A}\) allows you to express one trigonometric function in terms of another, which is an invaluable technique in both calculus and algebraic trigonometry. Once you familiarize yourself with these identities, they serve as powerful tools for simplifying complex trigonometric expressions and solving equations.

The Pythagorean Theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed algebraically as \(c^2 = a^2 + b^2\) where \(c\) represents the length of the hypotenuse, and \(a\) and \(b\) represent the lengths of the triangle's other two sides.

In trigonometry, the Pythagorean Theorem is linked with the Pythagorean identities. When dealing with trigonometric functions of acute angles, the theorem can be applied to a right triangle where \(c\) would be the hypotenuse (equivalent to 1 if considering a unit circle), \(a\) and \(b\) are the sides adjacent and opposite to the angle, respectively. This connection allows us to translate geometric information into trigonometric language and vice versa, enabling the solution of problems like the one provided in the step-by-step solution.

Solving trigonometric equations involves finding the values of the angles that satisfy the trigonometric expressions presented in the equation. These equations can be simple or complex, involving one or multiple trigonometric functions. The steps typically include isolating the trigonometric function, using trigonometric identities to simplify the equation, and then applying inverse trigonometric functions if necessary to find the angle solutions.

In the context of the given example, solving for \(\cos A\) required understanding the relationship between \(\tan A\) and \(\cos A\), and then employing the Pythagorean identity to express \(\sin A\) in terms of \(\cos A\). This equation was then manipulated algebraically to isolate \(\cos A\), and then taking the square root of both sides. Knowing the ranges and signs of the trigonometric functions within specific intervals, like how \(\cos A\) is positive in the interval \(0 < A < \frac{\pi}{2}\), is also crucial to arriving at the correct solution. Thus, solving trigonometric equations is a multi-step process that combines algebra with a deep understanding of trigonometry.