Question

Simplify the following expression using inverse trigonometric properties: \( \tan\left(\sin^{-1}x + \cos^{-1}x\right) \) where \( |x| \le 1 \).

• A. Not defined (\( \infty \))
• B. \( 1 \)
• C. \( 0 \)
• D. \( \frac{\pi}{2} \)

Hint:

Memorize the three core complementary inverse identities to save time on multiple-choice questions: \( \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \), \( \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \), and \( \sec^{-1}x + \csc^{-1}x = \frac{\pi}{2} \).

Answer 1

The Correct Option is A

Concept: Inverse trigonometric functions share complementary angle identity pairs. For any valid domain argument value where \( |x| \le 1 \), the sum of inverse sine and inverse cosine matches a constant right angle expression: \[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \]

Step 1: Apply the complementary identity property to substitute the interior expression.
Identify the core identity inside the function brackets: \[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \] Plugging this constant angle value into our outer function expression simplifies it to: \[ \tan\left(\sin^{-1}x + \cos^{-1}x\right) = \tan\left(\frac{\pi}{2}\right) \]

Step 2: Evaluate the outer trigonometric tangent function value.
The tangent of an angle is defined as the ratio of sine to cosine: \[ \tan\left(\frac{\pi}{2}\right) = \frac{\sin(\pi/2)}{\cos(\pi/2)} = \frac{1}{0} \] Because division by zero is impossible in real arithmetic, the value is not defined (approaching positive infinity).