If `x in (0,1)` and `f(x)=sec{tan^(-1)((sin(cos^(-1)x)+cos(sin^(-1)x))/(cos(cos^(-1)x)+sin(sin^(-1)x)))}`, then `sum_(r=2)^(10)f(1/r)` is _______ .

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the Function
Given the function:
\[ f(x) = \sec\left(\tan^{-1}\left(\frac{\sin(\cos^{-1}x) + \cos(\sin^{-1}x)}{\cos(\cos^{-1}x) + \sin(\sin^{-1}x)}\right)\right) \]

### Step 2: Simplify the Components
We know the following identities:
- \( \sin(\cos^{-1}x) = \sqrt{1 - x^2} \)
- \( \cos(\sin^{-1}x) = \sqrt{1 - x^2} \)
- \( \cos(\cos^{-1}x) = x \)
- \( \sin(\sin^{-1}x) = x \)

Using these identities, we can simplify the function:
- The numerator becomes:
\[ \sin(\cos^{-1}x) + \cos(\sin^{-1}x) = \sqrt{1 - x^2} + \sqrt{1 - x^2} = 2\sqrt{1 - x^2} \]
- The denominator becomes:
\[ \cos(\cos^{-1}x) + \sin(\sin^{-1}x) = x + x = 2x \]

Thus, we can rewrite \( f(x) \):
\[ f(x) = \sec\left(\tan^{-1}\left(\frac{2\sqrt{1 - x^2}}{2x}\right)\right) = \sec\left(\tan^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right)\right) \]

### Step 3: Use Right Triangle Relationships
In a right triangle where:
- Opposite side = \( \sqrt{1 - x^2} \)
- Adjacent side = \( x \)

The hypotenuse \( h \) can be calculated using the Pythagorean theorem:
\[ h = \sqrt{(\sqrt{1 - x^2})^2 + x^2} = \sqrt{1} = 1 \]

Thus, we have:
\[ \tan(\theta) = \frac{\sqrt{1 - x^2}}{x} \]
\[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{h}{x} = \frac{1}{x} \]

Therefore,
\[ f(x) = \frac{1}{x} \]

### Step 4: Calculate the Summation
Now we need to find:
\[ \sum_{r=2}^{10} f\left(\frac{1}{r}\right) \]

Substituting \( x = \frac{1}{r} \):
\[ f\left(\frac{1}{r}\right) = r \]

Thus, the summation becomes:
\[ \sum_{r=2}^{10} r \]

This is the sum of the first 10 natural numbers minus the sum of the first natural number:
\[ \sum_{r=1}^{10} r = \frac{10 \times 11}{2} = 55 \]
\[ \sum_{r=1}^{1} r = 1 \]

So,
\[ \sum_{r=2}^{10} r = 55 - 1 = 54 \]

### Final Answer
The value of \( \sum_{r=2}^{10} f\left(\frac{1}{r}\right) \) is:
\[ \boxed{54} \]