Let `tanalpha = a/b, alpha` being an acute angle. Denote by f(a,b) the value of the expression `a " cosec "2beta - b sec 2beta,` where `alpha = 6 beta`. Then `f(8,15)` is equal to …………….

Answer

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

### Step 1: Identify the given values
We are given that:
- \(\tan \alpha = \frac{a}{b}\)
- \(\alpha = 6\beta\)
- We need to find \(f(8, 15)\).

From this, we can identify:
- \(a = 8\)
- \(b = 15\)

### Step 2: Define the function \(f(a, b)\)
The function \(f(a, b)\) is defined as:
\[ f(a, b) = a \cdot \csc(2\beta) - b \cdot \sec(2\beta) \]

### Step 3: Express \(\csc(2\beta)\) and \(\sec(2\beta)\) in terms of \(\alpha\)
Using the definitions of sine and cosine:
\[ \csc(2\beta) = \frac{1}{\sin(2\beta)}, \quad \sec(2\beta) = \frac{1}{\cos(2\beta)} \]

### Step 4: Use the triangle relationships
From the right triangle with sides \(a\) and \(b\):
- The hypotenuse \(h\) can be calculated using the Pythagorean theorem:
\[ h = \sqrt{a^2 + b^2} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \]

### Step 5: Calculate \(\sin(\alpha)\) and \(\cos(\alpha)\)
Using the definitions:
\[ \sin(\alpha) = \frac{a}{h} = \frac{8}{17}, \quad \cos(\alpha) = \frac{b}{h} = \frac{15}{17} \]

### Step 6: Substitute into the function \(f(a, b)\)
Substituting \(\sin(2\beta)\) and \(\cos(2\beta)\):
\[ f(8, 15) = 8 \cdot \frac{1}{\sin(2\beta)} - 15 \cdot \frac{1}{\cos(2\beta)} \]
This can be rewritten as:
\[ f(8, 15) = \frac{8 \cdot \cos(2\beta) - 15 \cdot \sin(2\beta)}{\sin(2\beta) \cdot \cos(2\beta)} \]

### Step 7: Use the sine subtraction formula
Using the sine subtraction formula:
\[ \sin(6\beta - 2\beta) = \sin(4\beta) = \sin(6\beta)\cos(2\beta) - \cos(6\beta)\sin(2\beta) \]

### Step 8: Final expression for \(f(a, b)\)
Thus, we can express \(f(a, b)\) as:
\[ f(a, b) = \sqrt{a^2 + b^2} \cdot \frac{\sin(4\beta)}{\sin(2\beta) \cdot \cos(2\beta)} \]

### Step 9: Substitute values into the function
Substituting \(a = 8\) and \(b = 15\):
\[ f(8, 15) = 2 \cdot \sqrt{8^2 + 15^2} = 2 \cdot 17 = 34 \]

### Final Answer
Thus, \(f(8, 15) = 34\).

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