cos inverse(x^2/2 + square root of 1-x^2 * square root of 1- x^2/4)=cos inverse x/2 - cos inverse x holds for what value of x?

To solve the equation involving the cosine inverse function, we need to analyze the expression carefully. The equation is:

cos^-1(x²/2 + √(1 - x²) * √(1 - x²/4)) = cos^-1(x/2) - cos^-1(x).

First, let's simplify the left-hand side of the equation. The term inside the cosine inverse function can be rewritten. We know that:

• √(1 - x²) represents the sine of an angle when x is the cosine of that angle.
• √(1 - x²/4) can also be interpreted similarly.

Now, let's denote:

• y = cos^-1(x)
• Then, cos(y) = x.

Using this substitution, we can express the left-hand side as:

cos^-1(x²/2 + √(1 - x²) * √(1 - x²/4)) = cos^-1(x²/2 + (√(1 - x²) * √(1 - (x²/4)))

Next, we can simplify the term inside the cosine inverse. Let's calculate:

√(1 - x²/4) = √(4 - x²) / 2.

Now, substituting this back, we have:

cos^-1(x²/2 + (√(1 - x²) * (√(4 - x²

Next, we can analyze the right-hand side:

cos^-1(x/2) - cos^-1(x) can be rewritten using the cosine subtraction formula:

cos^-1(x/2) - cos^-1(x) = cos^-1(x/2 * x) = cos^-1(x²/2).

Now, we can equate both sides:

cos^-1(x²/2 + (√(1 - x²) * (√(4 - x^2-1(x²/2).

For the equality to hold, the arguments of the cosine inverse must be equal:

x²/2 + (√(1 - x²) * (√(4 - x²²/2.

Subtracting x²/2 from both sides gives us:

√(1 - x²) * (√(4 - x²

This implies that either √(1 - x²) = 0 or √(4 - x²

Solving these gives us:

• From √(1 - x²) = 0, we find x = ±1.
• From √(4 - x²

However, since x must lie within the range of the cosine function, which is [-1, 1], the only valid solution is:

x = 1.

Thus, the equation holds true for x = 1.