Find the value of cot `(pi)/(32)-"tan"(pi)/(32)-2"tan"(pi)/(16)`

Options

A. `4 "cot"(pi)/(8)`
B. 0
C. `2"cot"(pi)/(8)`
D. `"cot"(pi)/(8)`

Answer

To solve the problem \( \cot\left(\frac{\pi}{32}\right) - \tan\left(\frac{\pi}{32}\right) - 2\tan\left(\frac{\pi}{16}\right) \), we will follow these steps:

### Step 1: Define the angles
Let:
\[ \theta = \frac{\pi}{32} \]
Then:
\[ 2\theta = \frac{\pi}{16} \]

### Step 2: Rewrite the expression
The expression can be rewritten using our definition of \( \theta \):
\[ \cot(\theta) - \tan(\theta) - 2\tan(2\theta) \]

### Step 3: Use trigonometric identities
We know that:
\[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \quad \text{and} \quad \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
Thus, we can rewrite the expression:
\[ \frac{\cos(\theta)}{\sin(\theta)} - \frac{\sin(\theta)}{\cos(\theta)} - 2\tan(2\theta) \]

### Step 4: Combine the first two terms
Finding a common denominator for the first two terms:
\[ \frac{\cos^2(\theta) - \sin^2(\theta)}{\sin(\theta)\cos(\theta)} - 2\tan(2\theta) \]

### Step 5: Use the double angle formula
Using the double angle identity:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
We can express \( 2\tan(2\theta) \) as:
\[ 2\tan(2\theta) = \frac{4\tan(\theta)}{1 - \tan^2(\theta)} \]

### Step 6: Substitute back into the expression
Now we substitute back:
\[ \frac{\cos^2(\theta) - \sin^2(\theta)}{\sin(\theta)\cos(\theta)} - \frac{4\tan(\theta)}{1 - \tan^2(\theta)} \]

### Step 7: Simplify the expression
We can simplify the expression further by substituting \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \):
\[ \frac{\cos^2(\theta) - \sin^2(\theta)}{\sin(\theta)\cos(\theta)} - \frac{4\frac{\sin(\theta)}{\cos(\theta)}}{1 - \left(\frac{\sin(\theta)}{\cos(\theta)}\right)^2} \]

### Step 8: Final simplification
After simplifying, we find:
\[ \cot(4\theta) = \cot\left(\frac{\pi}{8}\right) \]

### Step 9: Conclusion
Thus, the final value of the expression is:
\[ \cot\left(\frac{\pi}{8}\right) \]