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The value of $ \sinh (3) $ $ - \cosh (3) $ $ = $
A. \[{e^{ - 3}}\]
B. \[ - {e^{ - 3}}\]
C. \[{e^3}\]
D. \[ - {e^3}\]

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Last updated date: 13th Jun 2024
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Answer
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Hint: Now, in this question hyperbolic trigonometric functions of $ \cos $ and $ \sin $ are mentioned. The hyperbolic functions are to be written in the exponential functions. Thereafter we will have to simplify the obtained equation to find the trigonometric equation’s value.
Formula used: We will have to use the formula of hyperbolic cos function, $ \cosh x $ $ = \dfrac{{{e^x} + {e^{ - x}}}}{2} $ and hyperbolic sine function, $ \sinh x $ $ = \dfrac{{{e^x} - {e^{ - x}}}}{2} $ .

Complete step-by-step answer:
According to the given information, we have
The first function is $ A = $ $ \sinh (3) $
The formula for $ \sinh x $ in terms of exponential function is,
 $ \sinh x $ $ = \dfrac{{{e^x} - {e^{ - x}}}}{2} $
Substituting x=3 in the above given equation we get,
 $ \sinh (3) $ \[ = \dfrac{{{e^3} - {e^{ - 3}}}}{2}\] $ ...(1) $
The first function is $ B = $ $ \cosh (3) $
The formula for $ \cosh x $ in terms of exponential function is,
 $ \cosh x $ $ = \dfrac{{{e^x} + {e^{ - x}}}}{2} $
Substituting x=3 in the above given equation we get,
 $ \cosh (3) $ $ = \dfrac{{{e^3} + {e^{ - 3}}}}{2} $ $ ...(2) $
According to the given data we have to calculate $ \sinh (3) $ $ - \cosh (3) $ , which is equal to,
 $ \sinh (3) $ $ - \cosh (3) $ \[ = \dfrac{{{e^3} - {e^{ - 3}}}}{2}\] $ - \dfrac{{{e^3} + {e^{ - 3}}}}{2} $
\[ \Rightarrow \dfrac{{{e^3} - {e^{ - 3}} - ({e^3} - {e^{ - 3}})}}{2}\]
\[ \Rightarrow \dfrac{{ - 2{e^{ - 3}}}}{2}\]\[ = - {e^{ - 3}}\]
So, the correct answer is “Option B”.

Note: In order to solve problems of this type the key is to have a basic understanding of trigonometric equations and values and also learn its implications. Hyperbolic functions are very similar to the trigonometric functions but are expressed in the form of exponential functions and the most common of them are $ \cosh x $ and $ \sinh x $ .
The formula for $ \cosh x $ in terms of exponential function is,
 $ \cosh x $ $ = \dfrac{{{e^x} + {e^{ - x}}}}{2} $
The formula for $ \sinh x $ in terms of exponential function is,
 $ \sinh x $ $ = \dfrac{{{e^x} - {e^{ - x}}}}{2} $