Question

The value of
\[ \sin \left[ \tan^{-1} \left( \frac{1 - x^2}{2x} \right) + \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right) \right] \]

• A. 0
• B. 1
• C. -1
• D. \( \frac{1}{2} \)

Hint:

When dealing with the sum of inverse trigonometric functions, you can use the identity for the sine of a sum to simplify the expression.

Answer 1

The Correct Option is B

Step 1: Simplify the trigonometric functions.
We are given a sum of two inverse trigonometric functions:
\[ \sin \left[ \tan^{-1} \left( \frac{1 - x^2}{2x} \right) + \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right) \right] \]
The first term inside the sine function is the inverse tangent, and the second term is the inverse cosine. To simplify this, we can use the identity for the sum of two inverse trigonometric functions:
\[ \sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B) \]
We need to compute the sine and cosine of each individual inverse trigonometric term.

Step 2: Work on the first term.
Let \( A = \tan^{-1} \left( \frac{1 - x^2}{2x} \right) \). For this term, we can use the identity: \[ \tan A = \frac{1 - x^2}{2x} \]
From this, we know that:
\[ \sin A = \frac{1 - x^2}{\sqrt{(1 - x^2)^2 + (2x)^2}} = \frac{1 - x^2}{\sqrt{1 - 2x^2 + x^4 + 4x^2}} = \frac{1 - x^2}{\sqrt{1 + 2x^2 + x^4}} \]
And similarly for \( \cos A \), using:
\[ \cos A = \frac{2x}{\sqrt{1 + 2x^2 + x^4}} \]

Step 3: Work on the second term.
For the second term \( B = \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right) \), we know that:
\[ \cos B = \frac{1 - x^2}{1 + x^2} \]
From the Pythagorean identity \( \sin^2 B + \cos^2 B = 1 \), we can calculate \( \sin B \) as: \[ \sin B = \sqrt{1 - \left( \frac{1 - x^2}{1 + x^2} \right)^2} \]

Step 4: Apply the sum formula.
Now, we apply the sum identity:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
Substitute the expressions for \( \sin A \), \( \cos A \), \( \sin B \), and \( \cos B \) into this formula.

Step 5: Simplify the result.
After simplification, the terms involving \( x \) will cancel out, and the result will simplify to:
\[ \boxed{1} \]

Step 6: Conclusion.
Thus, the value of the given expression is \( \boxed{1} \).