Answer
$$csch^{-1}\Big(-\frac{1}{\sqrt3}\Big)=\ln(2-\sqrt3)$$
Work Step by Step
We have $$csch^{-1}x=\ln\Big(\frac{1}{x}+\frac{\sqrt{1+x^2}}{|x|}\Big)$$ for $x\ne0$
Therefore, $$csch^{-1}\Big(-\frac{1}{\sqrt3}\Big)=\ln\Big(\frac{1}{-\frac{1}{\sqrt3}}+\frac{\sqrt{1+\frac{1}{3}}}{\frac{1}{\sqrt3}}\Big)$$ $$=\ln\Big(-\sqrt3+\frac{\sqrt{\frac{4}{3}}}{\frac{1}{\sqrt3}}\Big)=\ln\Big(-\sqrt3+\frac{\frac{2}{\sqrt3}}{\frac{1}{\sqrt3}}\Big)$$ $$=\ln\Big(-\sqrt3+\frac{2}{1}\Big)=\ln(2-\sqrt3)$$
