If $ cos^{-1}\sqrt{p} + cos^{-1} \sqrt{1-p} +cos^{-1} \sqrt{1-q} =\frac{3\pi}{4}, $ then the value of q, is
Answer & explanation
Correct answer: option 3
Clearly, left hand side of the given equation is meaningful if $0 ≤ p ≤ 1$ and $0 ≤ q ≤ 1$.
Let $ p = cos^2\theta .$ Then,
$ cos^{-1}\sqrt{p} + cos^{-1} \sqrt{1-p} +cos^{-1} \sqrt{1-q} =\frac{3\pi}{4}$
$⇒ cos^{-1}(cos \theta ) + cos^{-1} (sin \theta ) + cos^{-1} \sqrt{1-q} =\frac{3\pi}{4}$
$⇒ \theta + \frac{\pi}{2} - \theta + cos^{-1} \sqrt{1-q} =\frac{3\pi}{4}$
$⇒cos^{-1} \sqrt{1-q} =\frac{\pi}{4}$
$⇒ \sqrt{1-q} =\frac{1}{\sqrt{2}} ⇒ 1 - q = \frac{1}{2} ⇒ q = \frac{1}{2}$