Statement I `tan alpha+2tan 2 alpha+4 tan 4 alpha + 8 tan 8 alpha+16 cot 16 alpha=cot alpha`
Statement II `cot alpha- tan alpha=2 cot 2 alpha`

Options

A. Both Statement I and Statement II are individually true and R is the correct explanation of Statement I.
B. Both Statement I and Statement II are individually true but Statement II is not the correct explanaton of Statement I.
C. Statement I is true but Statement II is false.
D. Statement I is false but Statement II is true.

Answer

To solve the given statements, we will analyze each statement step by step.

### Step 1: Analyze Statement I
We need to prove that:
\[ \tan \alpha + 2 \tan 2\alpha + 4 \tan 4\alpha + 8 \tan 8\alpha + 16 \cot 16\alpha = \cot \alpha \]

### Step 2: Rewrite Cotangent
We know that:
\[ \cot 16\alpha = \frac{1}{\tan 16\alpha} \]
Thus, we can rewrite the equation as:
\[ \tan \alpha + 2 \tan 2\alpha + 4 \tan 4\alpha + 8 \tan 8\alpha + \frac{16}{\tan 16\alpha} = \cot \alpha \]

### Step 3: Use the Identity for Cotangent
Using the identity for cotangent:
\[ \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \]
We can express \( \cot \alpha \) in terms of \( \tan \alpha \):
\[ \cot \alpha = \frac{1}{\tan \alpha} \]

### Step 4: Analyze Statement II
We need to prove that:
\[ \cot \alpha - \tan \alpha = 2 \cot 2\alpha \]

### Step 5: Rewrite Cotangent for 2α
Using the double angle formula for cotangent:
\[ \cot 2\alpha = \frac{\cot^2 \alpha - 1}{2 \cot \alpha} \]
Thus:
\[ 2 \cot 2\alpha = \frac{2(\cot^2 \alpha - 1)}{2 \cot \alpha} = \frac{\cot^2 \alpha - 1}{\cot \alpha} \]

### Step 6: Substitute in Statement II
Substituting \( 2 \cot 2\alpha \) back into Statement II gives:
\[ \cot \alpha - \tan \alpha = \frac{\cot^2 \alpha - 1}{\cot \alpha} \]

### Step 7: Verify the Equality
To verify the equality:
1. Rewrite \( \tan \alpha \) as \( \frac{1}{\cot \alpha} \).
2. Substitute into the equation:
\[ \cot \alpha - \frac{1}{\cot \alpha} = \frac{\cot^2 \alpha - 1}{\cot \alpha} \]

### Step 8: Simplify
Multiply through by \( \cot \alpha \):
\[ \cot^2 \alpha - 1 = \cot^2 \alpha - 1 \]
This confirms that Statement II is true.

### Conclusion
Both statements are true, and Statement II serves as a valid explanation for Statement I.