Lat `x=2^((log_(2)3) (log_(3) 4)....)`, where the last term in exponent is `log_(99) 100`, then the value of `cos ((xpi)/4)+sin ((x pi)/4)` is equal to -

Options

A. `1`
B. `1/sqrt(2)`
C. `sqrt(2)`
D. `-1`

Answer

To solve the problem step by step, we start with the expression given for \( x \):

### Step 1: Define the expression for \( x \)
We have:
\[ x = 2^{(\log_2 3)(\log_3 4)(\log_4 5) \ldots (\log_{99} 100)} \]

### Step 2: Rewrite the logarithmic terms
Using the change of base formula for logarithms, we can express each term as:
\[ \log_a b = \frac{\log b}{\log a} \]
Thus, we can rewrite \( x \) as:
\[ x = 2^{\left(\frac{\log 3}{\log 2}\right) \left(\frac{\log 4}{\log 3}\right) \left(\frac{\log 5}{\log 4}\right) \ldots \left(\frac{\log 100}{\log 99}\right)} \]

### Step 3: Simplify the product of logarithms
Notice that in the product, all intermediate terms will cancel out:
\[ x = 2^{\frac{\log 100}{\log 2}} \]
This is because all the logs in the numerator and denominator cancel each other out.

### Step 4: Simplify further
Now we can simplify \( x \):
\[ x = 2^{\log_2 100} \]
Using the property of logarithms, we know that \( a^{\log_a b} = b \). Therefore:
\[ x = 100 \]

### Step 5: Substitute \( x \) into the expression
Now we need to find the value of:
\[ \cos\left(\frac{x \pi}{4}\right) + \sin\left(\frac{x \pi}{4}\right) \]
Substituting \( x = 100 \):
\[ \cos\left(\frac{100 \pi}{4}\right) + \sin\left(\frac{100 \pi}{4}\right) = \cos(25\pi) + \sin(25\pi) \]

### Step 6: Evaluate \( \cos(25\pi) \) and \( \sin(25\pi) \)
We know that:
- \( \cos(25\pi) = -1 \) (since \( 25 \) is an odd integer)
- \( \sin(25\pi) = 0 \)

### Step 7: Combine the results
Thus:
\[ \cos(25\pi) + \sin(25\pi) = -1 + 0 = -1 \]

### Final Answer
The value of \( \cos\left(\frac{x \pi}{4}\right) + \sin\left(\frac{x \pi}{4}\right) \) is:
\[ \boxed{-1} \]