Explanation
To solve this, we apply a specific property of inverse trigonometric functions that relates sec−1(x) to cos−1(x).
1. Applying the Reciprocal Property
We know that for any value in the domain:
Applying this to the first term of our equation:
sec−1(x−1x+1)=cos−1(x+1x−1)
2. Substituting back into the Equation
Now, the original expression for y becomes:
y=cos−1(x+1x−1)+sin−1(x+1x−1)
3. Using the Identity Property
Recall the fundamental identity for inverse trigonometric functions:
By letting A=x+1x−1, we can see that:
4. Differentiation
Since y is now simplified to a constant value (2π), we differentiate both sides with respect to x:
The derivative of any constant is zero:
Correct Option: (c) 0
