Question

Evaluate \[ \left[ \sin \left( \tan^{-1} \frac{3}{4} \right) \right]^2 + \left[ \sin \left( \tan^{-1} \frac{4}{3} \right) \right]^2 \]

• A. 5
• B. 1
• C. -1
• D. 0

Hint:

When dealing with trigonometric functions of inverse trigonometric functions, use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to simplify the expression.

Answer 1

The Correct Option is B

Step 1: Simplify the terms.
We start by simplifying the individual terms. For \( \sin \left( \tan^{-1} \frac{3}{4} \right) \), using the identity \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \), we find that \( \sin \left( \tan^{-1} \frac{3}{4} \right) = \frac{3}{5} \). Similarly, for \( \sin \left( \tan^{-1} \frac{4}{3} \right) \), we find that \( \sin \left( \tan^{-1} \frac{4}{3} \right) = \frac{4}{5} \).

Step 2: Apply the Pythagorean identity.
Now, squaring both terms: \[ \left( \frac{3}{5} \right)^2 + \left( \frac{4}{5} \right)^2 = \frac{9}{25} + \frac{16}{25} = 1. \]
Step 3: Conclusion.
Thus, the correct answer is 1, corresponding to option (B).