
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


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In the field of engineering, hyperbolic sine function $shx = \frac{e^{x}-e^{-x}}{2}$ and hyperbolic cosine function $chx = \frac{e^{x}+e^{-x}}{2}$ are often used. In fact, the hyperbolic sine and cosine functions are similar to the sine and cosine functions we have learned. For example, the identity $\cos (x+y)=\cos x\cos y-\sin x\sin y$ holds for sine and cosine functions, while the identity $cb(x+y)=chxchy+shxshy$ holds for hyperbolic sine and cosine functions. Please derive a new relationship between hyperbolic sine and cosine functions by analogy with the relationships between sine and cosine functions.

Present the answer in LaTex format: \boxed{Your answer}
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