The value of `cot36^(@)cot72^(@),` is

Options

A. `1//5`
B. `1//sqrt5`
C. 1
D. `1//3`

Answer

To find the value of \( \cot 36^\circ \cot 72^\circ \), we can follow these steps:

### Step 1: Rewrite cotangent in terms of tangent
We know that:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Thus, we can rewrite the expression:
\[ \cot 36^\circ \cot 72^\circ = \frac{1}{\tan 36^\circ} \cdot \frac{1}{\tan 72^\circ} = \frac{1}{\tan 36^\circ \tan 72^\circ} \]

### Step 2: Use the identity for tangent of complementary angles
We know that:
\[ \tan(90^\circ - \theta) = \cot \theta \]
So, we can express \( \tan 72^\circ \) as:
\[ \tan 72^\circ = \cot 18^\circ \]
Thus, we have:
\[ \cot 36^\circ \cot 72^\circ = \frac{1}{\tan 36^\circ \cot 18^\circ} = \frac{1}{\tan 36^\circ \cdot \frac{1}{\tan 18^\circ}} = \frac{\tan 18^\circ}{\tan 36^\circ} \]

### Step 3: Use the double angle formula for tangent
Using the double angle formula:
\[ \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \]
Let \( x = 18^\circ \), then:
\[ \tan 36^\circ = \tan(2 \cdot 18^\circ) = \frac{2 \tan 18^\circ}{1 - \tan^2 18^\circ} \]
Substituting this into our expression gives:
\[ \cot 36^\circ \cot 72^\circ = \frac{\tan 18^\circ}{\frac{2 \tan 18^\circ}{1 - \tan^2 18^\circ}} = \frac{1 - \tan^2 18^\circ}{2} \]

### Step 4: Substitute the value of \( \tan 18^\circ \)
We know:
\[ \tan 18^\circ = \sqrt{5 - 2\sqrt{5}}/5 \]
Substituting this value into our expression:
\[ \cot 36^\circ \cot 72^\circ = \frac{1 - \left(\frac{\sqrt{5 - 2\sqrt{5}}}{5}\right)^2}{2} \]

### Step 5: Simplify the expression
Calculating \( \left(\frac{\sqrt{5 - 2\sqrt{5}}}{5}\right)^2 \):
\[ = \frac{5 - 2\sqrt{5}}{25} \]
Thus, we have:
\[ \cot 36^\circ \cot 72^\circ = \frac{1 - \frac{5 - 2\sqrt{5}}{25}}{2} \]
Finding a common denominator:
\[ = \frac{\frac{25 - (5 - 2\sqrt{5})}{25}}{2} = \frac{\frac{20 + 2\sqrt{5}}{25}}{2} = \frac{20 + 2\sqrt{5}}{50} = \frac{2(10 + \sqrt{5})}{50} = \frac{10 + \sqrt{5}}{25} \]

### Final Result
Thus, the value of \( \cot 36^\circ \cot 72^\circ \) is:
\[ \frac{10 + \sqrt{5}}{25} \]