Question

$\left[-\frac{1}{Sinh^{2}a} + \frac{1}{Sinh^{4}a} - \frac{1}{2} \right]$

Answer 1

$\text{Coth}^4 a - 3\text{Coth}^2 a + \frac{3}{2}$

Answer 2

To simplify the given expression $\left[-\frac{1}{\text{Sinh}^{2}a} + \frac{1}{\text{Sinh}^{4}a} - \frac{1}{2} \right]$, we can use the fundamental hyperbolic identity.

The identity we will use is: $\text{Coth}^2 a - \text{Cosech}^2 a = 1$

Since $\text{Cosech}^2 a = \frac{1}{\text{Sinh}^2 a}$, we can write: $\text{Coth}^2 a - \frac{1}{\text{Sinh}^2 a} = 1$

From this, we can express $\frac{1}{\text{Sinh}^2 a}$ in terms of $\text{Coth}^2 a$: $\frac{1}{\text{Sinh}^2 a} = \text{Coth}^2 a - 1$

Let's substitute $x = \frac{1}{\text{Sinh}^2 a}$ into the given expression. The expression becomes: $x^2 - x - \frac{1}{2}$

Now, substitute $x = \text{Coth}^2 a - 1$ into this quadratic expression: $(\text{Coth}^2 a - 1)^2 - (\text{Coth}^2 a - 1) - \frac{1}{2}$

Expand the terms: $= (\text{Coth}^4 a - 2\text{Coth}^2 a + 1) - (\text{Coth}^2 a - 1) - \frac{1}{2}$ $= \text{Coth}^4 a - 2\text{Coth}^2 a + 1 - \text{Coth}^2 a + 1 - \frac{1}{2}$

Combine like terms: $= \text{Coth}^4 a - (2\text{Coth}^2 a + \text{Coth}^2 a) + (1 + 1 - \frac{1}{2})$ $= \text{Coth}^4 a - 3\text{Coth}^2 a + (2 - \frac{1}{2})$ $= \text{Coth}^4 a - 3\text{Coth}^2 a + \frac{4-1}{2}$ $= \text{Coth}^4 a - 3\text{Coth}^2 a + \frac{3}{2}$

This is the simplified form of the expression.