If ^ -1 p + ^ -1 1-p + ^ -1 1-q =(3 π)/(4) , then q

$$\begin{aligned} & \cos ^{-1} \sqrt{\mathrm{p}}-\cos ^{-1} \sqrt{1-\mathrm{p}}+\cos ^{-1} \sqrt{1-\mathrm{q}}=\frac{3 \pi}{4} \\ & \text { Let } \mathrm{t}=\cos ^{-1} \sqrt{\mathrm{p}} \\ & \Rightarrow \mathrm{p}=\cos ^2 \mathrm{t} \\ & \Rightarrow \mathrm{p}=1-\sin ^2 \mathrm{t} \\ & \Rightarrow \sin \mathrm{t}=\sqrt{1-\mathrm{p}} \\ & \Rightarrow \mathrm{t}=\sin ^{-1} \sqrt{1-\mathrm{p}}\end{aligned}$$\Rightarrow \cos ^{-1} \sqrt{\mathrm{p}}=\sin ^{-1} \sqrt{1-\mathrm{p}}\therefore \quad$Given equation becomes$\sin ^{-1} \sqrt{1-p}-\cos ^{-1} \sqrt{1-p}+\cos ^{-1} \sqrt{1-q}=\frac{3 \pi}{4}\therefore \quad \frac{\pi}{2}+\cos ^{-1} \sqrt{1-q}=\frac{3 \pi}{4} \quad \ldots\left[\because \cos ^{-1}a+\sin ^{-1} a=\frac{\pi}{2}\right]$$\begin{array}{ll}\therefore& \cos ^{-1} \sqrt{1-q}=\frac{-\pi}{4}\\ \therefore & \sqrt{1-q}=\cos \left(-\frac{\pi}{4}\right)\\ \therefore \text{ and } q=1-\frac{1}{2}\\ \therefore \text{ and } q=\frac{1}{2}\end{array}$$$