Assertion (A) : `(3^2)/(2!) +(3^4)/(4!)+(3^6)/(4!)+ .... =` Cosh 3 -1
Reason (R) : Cosh `x =1 +(x^2)/(2!) +x^4/(4!) + .....`

Options

A. A is true, R is true and R is correct explanation of A
B. A is true, R is true and R is not correct explanation of A
C. A is true , R is false
D. A is false, R is true

Answer

To solve the given assertion and reason, we will analyze the series and relate it to the hyperbolic cosine function.

### Step 1: Understand the series in the assertion
The assertion states:
\[ \frac{3^2}{2!} + \frac{3^4}{4!} + \frac{3^6}{6!} + \ldots \]
This series includes terms of the form \(\frac{3^{2n}}{(2n)!}\) for \(n = 1, 2, 3, \ldots\).

### Step 2: Relate the series to the hyperbolic cosine function
The hyperbolic cosine function, \(\cosh(x)\), is defined as:
\[ \cosh(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \ldots \]
If we replace \(x\) with \(3\), we get:
\[ \cosh(3) = 1 + \frac{3^2}{2!} + \frac{3^4}{4!} + \frac{3^6}{6!} + \ldots \]

### Step 3: Rearranging the equation
From the definition of \(\cosh(3)\), we can rearrange it to isolate the series:
\[ \cosh(3) - 1 = \frac{3^2}{2!} + \frac{3^4}{4!} + \frac{3^6}{6!} + \ldots \]
This shows that the series in the assertion is equal to \(\cosh(3) - 1\).

### Step 4: Conclusion
Thus, we can conclude that the assertion is true:
\[ \frac{3^2}{2!} + \frac{3^4}{4!} + \frac{3^6}{6!} + \ldots = \cosh(3) - 1 \]
The reason provided is also true, as it accurately describes the series expansion of \(\cosh(x)\).

### Final Answer
Both the assertion (A) and the reason (R) are true, and R is a correct explanation for A.

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