`"cosec"[2 cot^(-1)(5) + cos^(-1)(4/5)]` is equal to

Options

A. `56/33`
B. `65/56`
C. `65/33`
D. `75/56`

Answer

To solve the expression \( \csc[2 \cot^{-1}(5) + \cos^{-1}(\frac{4}{5})] \), we will follow these steps:

### Step 1: Let \( \theta = \cot^{-1}(5) \)
From the definition of cotangent, we know that:
\[ \cot(\theta) = 5 \implies \tan(\theta) = \frac{1}{5} \]
This can be represented in a right triangle where the adjacent side is 5 and the opposite side is 1.

### Step 2: Find the hypotenuse
Using the Pythagorean theorem:
\[ \text{Hypotenuse} = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26} \]

### Step 3: Find \( \sin(\theta) \) and \( \cos(\theta) \)
Now we can find:
\[ \sin(\theta) = \frac{1}{\sqrt{26}}, \quad \cos(\theta) = \frac{5}{\sqrt{26}} \]

### Step 4: Calculate \( 2\theta \)
Using the double angle formulas:
\[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) = 2 \cdot \frac{1}{\sqrt{26}} \cdot \frac{5}{\sqrt{26}} = \frac{10}{26} = \frac{5}{13} \]
\[ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = \left(\frac{5}{\sqrt{26}}\right)^2 - \left(\frac{1}{\sqrt{26}}\right)^2 = \frac{25}{26} - \frac{1}{26} = \frac{24}{26} = \frac{12}{13} \]

### Step 5: Find \( \cos^{-1}(\frac{4}{5}) \)
Let \( \phi = \cos^{-1}(\frac{4}{5}) \). Then:
\[ \cos(\phi) = \frac{4}{5} \]
Using the Pythagorean identity, we find:
\[ \sin(\phi) = \sqrt{1 - \cos^2(\phi)} = \sqrt{1 - \left(\frac{4}{5}\right)^2} = \sqrt{1 - \frac{16}{25}} = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

### Step 6: Combine angles
Now we need to find \( \csc(2\theta + \phi) \):
\[ \sin(2\theta + \phi) = \sin(2\theta)\cos(\phi) + \cos(2\theta)\sin(\phi) \]
Substituting the values:
\[ \sin(2\theta + \phi) = \left(\frac{5}{13}\right) \left(\frac{4}{5}\right) + \left(\frac{12}{13}\right) \left(\frac{3}{5}\right) \]
Calculating each term:
\[ = \frac{20}{65} + \frac{36}{65} = \frac{56}{65} \]

### Step 7: Find \( \csc(2\theta + \phi) \)
Finally, we find:
\[ \csc(2\theta + \phi) = \frac{1}{\sin(2\theta + \phi)} = \frac{1}{\frac{56}{65}} = \frac{65}{56} \]

Thus, the final answer is:
\[ \csc[2 \cot^{-1}(5) + \cos^{-1}(\frac{4}{5})] = \frac{65}{56} \]

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