Find the solutions of the equation cos(cos⁻¹x) = cos⁻¹(cos⁻¹x).

To solve the equation cos(cos^-1 x) = cos^-1(cos^-1 x), we can break it down step by step.

Understanding the Components

The left side, cos(cos^-1 x), simplifies directly to x because the cosine function and its inverse cancel each other out.

Analyzing the Right Side

The right side, cos^-1(cos^-1 x), represents the angle whose cosine is cos^-1 x. This simplifies to cos^-1 x itself, as it is the output of the inverse cosine function.

Setting Up the Equation

Now, we can rewrite the original equation:

• x = cos^-1 x

Finding Solutions

To find the values of x that satisfy this equation, we can analyze the function y = cos^-1 x. The range of cos^-1 x is from 0 to π (approximately 3.14), and it is defined for -1 ≤ x ≤ 1.

Graphical Interpretation

Graphing y = x and y = cos^-1 x helps us find the intersection points. The two functions intersect at:

• x = 0
• x = 1

Final Solutions

The solutions to the equation cos(cos^-1 x) = cos^-1(cos^-1 x) are:

• x = 0
• x = 1