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The cosine inverse function, denoted as \( \text {cos}^{-1} x \), is one of the inverse trigonometric functions. It helps to find the angle whose cosine is the given number. For example: \( \text {cos}^{-1} \left ( -1 \right ) = \pi \) means that the angle whose cosine is -1 is \pi \ radian.
Inverse trigonometric functions only provide solutions within a restricted range. For cosine inverse, the possible output angles lie between 0 and \pi \ radians. This function is useful in solving various trigonometric equations by transforming an equation involving trigonometric functions into a more straightforward algebraic equation.
To solve the equation \(4 \text {cos}^{-1} x - 2\pi = 2 \text {cos}^{-1} x \), we can isolate \smash{x}\ by using the properties and definitions of the cosine inverse function.

Solving trigonometric equations often involves using inverse trigonometric functions. These functions aid in finding angles that satisfy the given trigonometric conditions. Here's a simple strategy for solving such equations:
- Isolate the trigonometric expression.
- Apply the inverse trigonometric function to both sides.
- Solve for the variable of interest.
In our specific example, the equation \(4 \text {cos}^{-1} x - 2\pi = 2 \text {cos}^{-1} x \) can be simplified and rearranged. Let's subtract \smash{2 \text {cos}^{-1} x}\ from both sides to isolate the trigonometric expression:
\(4 \text {cos}^{-1} x - 2\pi - 2 \text {cos}^{-1} x = 2 \text {cos}^{-1} x - 2 \text {cos}^{-1} x \)
This simplifies to:
\(2 \text {cos}^{-1} x - 2\pi = 0 \)
Adding \pi \ to each side delivers:
\(2 \text {cos}^{-1} x = 2\pi \)
Dividing both sides by 2, we find:
\( \text {cos}^{-1} x = \pi \)
Thus, \( \text {cos}^{-1} x = \pi \). This tells us the angle which has a cosine value equal to \smash{x}\.

Finding exact solutions in trigonometry often involves using known values of trigonometric functions at specific angles. The cosine function has specific angles where its values are well-known. For example:
- \smash {\text {cos} \(0 \) = 1}
- \smash {\text {cos} \( \frac {\pi}{2} \) = 0}
- \smash {\text {cos} \left ( \pi \right ) = -1}
In our equation, we identified \smash { \text {cos}^{-1} x = \pi }\. By recognizing that \smash { \cos \left ( \pi \right ) = -1 }\, we find that \smash{x = -1}\. Therefore, \smash { x } equals \smash { -1 }\, solving the original trigonometric equation.
Using the properties of the inverse trigonometric functions and exact trigonometric values simplifies complex problems. This approach helps in obtaining a clear and accurate solution to trigonometric equations.