Hyperbolic Functions

Given: . We want to find .

First, recall that . Thus, .

Substitute :

.

Now, compute .

We utilize the triple angle formula for :

.

Substitute :

,

,

.

Calculate:

.

Simplify the terms:

, ,

.

Thus, .

However, this is incorrect. The correct simplification reveals:

.

The correct answer is .

We are given the following equations:

We are tasked with finding .

Step 1: Express and in terms of hyperbolic functions.

We start by recalling the definitions of inverse hyperbolic functions:

• means .
• means .

From the given equation , we can deduce:

Using the identity for , we have:

Thus, .

Next, from the equation , we can deduce:

Using the identity for , we have:

Thus:

Therefore, .

Step 2: Calculate .

Now that we have and , we compute:

To subtract these fractions, we first find a common denominator:

Thus, .

Step 3: Compute .

We are asked to find , which is . Recall that:

Applying this formula:

Final Answer:

The final answer is .