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How to derive the formulas for hyperbolic cosecant?

The hyperbolic cosecant function is defined as csch(x) = 1/sinh(x). To derive its formulas, we need to use the definitions of the hyperbolic sine and cosine functions.

First, we know that sinh(x) = (e^x - e^(-x))/2 and cosh(x) = (e^x + e^(-x))/2. Therefore, csch(x) = 1/sinh(x) = 2/(e^x - e^(-x)).

To simplify this expression, we can multiply the numerator and denominator by e^x, giving csch(x) = 2e^x/(e^(2x) - 1). We can also write this as csch(x) = -2e^(-x)/(1 - e^(-2x)).

Using these formulas, we can find the derivatives of csch(x) with respect to x. Using the quotient rule, we have:

d/dx csch(x) = -2e^(-x)/(1 - e^(-2x))^2 * (-2e^(-x)) = 2csch(x)coth(x)

Similarly, we can find the integral of csch(x) with respect to x. Using the substitution u = e^x, du = e^x dx, we have:

∫csch(x) dx = ∫(1/u) * (1/(1 - u^2)) du = -1/2 ln|(u - 1)/(u + 1)| + C

= -1/2 ln|(e^x - 1)/(e^x + 1)| + C

Therefore, the formulas for hyperbolic cosecant are csch(x) = 2/(e^x - e^(-x)) = -2e^(-x)/(1 - e^(-2x)), d/dx csch(x) = 2csch(x)coth(x), and ∫csch(x) dx = -1/2 ln|(e^x - 1)/(e^x + 1)| + C.

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