If sinh ^{-1}(-sqrt{3})+cosh ^{-1}(2)=K, then | vedclass

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(D) Given: $\sinh ^{-1}(-\sqrt{3})+\cosh ^{-1}(2)=K$ Let $x=\sinh ^{-1}(-\sqrt{3})$ and $y=\cosh ^{-1}(2)$. Then $x+y=K$. From the definitions, $\sinh

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(D) Given: $\sinh ^{-1}(-\sqrt{3})+\cosh ^{-1}(2)=K$ Let $x=\sinh ^{-1}(-\sqrt{3})$ and $y=\cosh ^{-1}(2)$. Then $x+y=K$. From the definitions, $\sinh x = -\sqrt{3}$ and $\cosh y = 2$. Using the identity $\cosh^2 x - \sinh^2 x = 1$, we have $\cosh^2 x = 1 + (-\sqrt{3})^2 = 1 + 3 = 4$. Since $\cosh x \geq 1$, we get $\cosh x = 2$. Using the identity $\cosh^2 y - \sinh^2 y = 1$, we have $2^2 - \sinh^2 y = 1$, so $\sinh^2 y = 3$. Since $\cosh^{-1}(2)$ is positive, $\sinh y = \sqrt{3}$. Now, $\cosh K = \cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y$. Substituting the values: $\cosh K = (2)(2) + (-\sqrt{3})(\sqrt{3}) = 4 - 3 = 1$.

If $\sinh ^{-1}(-\sqrt{3})+\cosh ^{-1}(2)=K$, then $\cosh K=$

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