Question:

<div class="solution"><p>\(\cos^{-1}\!\sqrt{p}+\sin^{-1}\!\sqrt{p}=\pi/2\) (complementary pair since \(\cos^{-1}\!\sqrt{1-p}=\sin^{-1}\!\sqrt{p}\)). So \(\pi/2+\cos^{-1}\!\sqrt{1-q}=3\pi/4\implies\cos^{-1}\!\sqrt{1-q}=\pi/4\implies\sqrt{1-q}=1/\sqrt{2}\implies q=1/2\).</p><p><strong>Answer: (2) 1/2</strong></p><div class="trap-box"><strong>Trap:</strong> Trying to solve for p as well -- the p-terms collapse immediately.<div class="key-concept"><strong>Key Concept:</strong> Complementary pair cos⁻^1\sqrtp + cos⁻^1\sqrt(1-p) = \pi/2