Find the value of sin⁻¹( cos⁻¹( sin⁻¹( cos⁻¹(-1)x) ) + cos⁻¹( sin( cos⁻¹(-1)x) ) )

• A: π/4
• B: π/2
• C: 3π/4
• D: 0

To solve the expression sin⁻¹( cos⁻¹( sin⁻¹( cos⁻¹(-1)x) ) + cos⁻¹( sin( cos⁻¹(-1)x) ), we need to break it down step by step.

Step 1: Evaluate cos⁻¹(-1)x

The value of cos⁻¹(-1) is π. Therefore, cos⁻¹(-1)x = πx.

Step 2: Find sin⁻¹(πx)

Next, we need to evaluate sin⁻¹(πx). However, the range of sin⁻¹ is limited to [-π/2, π/2]. For this expression to be valid, we need to ensure that πx falls within this range.

Step 3: Evaluate cos⁻¹(sin(πx))

Since sin(πx) will also depend on the value of x, we can say that cos⁻¹(sin(πx)) will yield a value based on the sine function's output.

Combining the Results

Now, we combine the results from the two evaluations:

• First part: sin⁻¹(πx)
• Second part: cos⁻¹(sin(πx))

Final Evaluation

Adding these two results together will yield a specific angle. After careful consideration, we find that the overall expression simplifies to π/2.

Final Answer

The value of the original expression is π/2, which corresponds to option B.