Maths · Inverse trigonometrical functions and their properties
Find the value of
Find the value of \( \sin ^{-1} x+\sin ^{-1} \frac{1}{x}+ \) \( \cos ^{-1} x+\cos ^{-1} \frac{1}{x} \)
- A. \( -\pi \)
- B. \( +\pi \)
- C. -2 \pi \)
- D. \( +2 \pi \)
Step-by-step solution
Using the identity sin^{-1} x + cos^{-1} x = π/2 for |x| ≤ 1, and similarly sin^{-1}(1/x) + cos^{-1}(1/x) = π/2 for |1/x| ≤ 1, the given sum equals π/2 + π/2 = π, provided both conditions hold, which occurs only when |x| = 1. At x = ±1, the sum evaluates to π. Hence, the value is +π.
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