If $tan 8 heta = cot 2 heta, where 0

Q: If $tan 8\theta = cot 2\theta, where 0 < 8\theta < \frac{\pi}{2}$, then what is the value of $tan 5\theta$

To solve the problem, we need to analyze the given equation and find the value of $ \tan 5\theta $. Given: \[ \tan 8\theta = \cot 2\theta \] We know that: \[ \cot 2\theta = \frac{1}{\tan 2\theta} \] Thus, the equation becomes: \[ \tan 8\theta = \frac{1}{\tan 2\theta} \] This implies: \[ \tan 8\theta \cdot \tan 2\theta = 1 \] Now, let's use the identity for tangent: \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] Since $ \tan 8\theta \cdot \tan 2\theta = 1 $, we can infer that: \[ 8\theta = 90^\circ - 2\theta \] Solving for $ \theta $: \[ 8\theta + 2\theta = 90^\circ \] \[ 10\theta = 90^\circ \] \[ \theta = 9^\circ \] Now, we need to find $ \tan 5\theta $: \[ \tan 5\theta = \tan(5 \times 9^\circ) = \tan 45^\circ \] We know that: \[ \tan 45^\circ = 1 \] Therefore, the value of $ \tan 5\theta $ is 1, which corresponds to Option 2. ### Explanation for Other Options: 1. **Option 1: $ \frac{1}{\sqrt{3}} $** - This value does not correspond to $ \tan 45^\circ $. It is the value of $ \tan 30^\circ $, which is not applicable here. 2. **Option 3: $ \sqrt{3} $** - This value corresponds to $ \tan 60^\circ $, which is not the angle we are dealing with in this problem. 3. **Option 4: 0** - This value corresponds to $ \tan 0^\circ $, which is not relevant to the given problem as $ \tan 5\theta = \tan 45^\circ $. Thus, the correct answer is indeed Option 2, as $ \tan 5\theta = 1 $.

Question

If $tan 8	heta = cot 2	heta, where 0 < 8	heta < \frac{\pi}{2}$, then what is the value of $tan 5	heta$

Accepted Answer

To solve the problem, we need to analyze the given equation and find the value of $ \tan 5\theta $.

Given:
$$ \tan 8\theta = \cot 2\theta $$

We know that:
$$ \cot 2\theta = \frac{1}{\tan 2\theta} $$

Thus, the equation becomes:
$$ \tan 8\theta = \frac{1}{\tan 2\theta} $$

This implies:
$$ \tan 8\theta \cdot \tan 2\theta = 1 $$

Now, let's use the identity for tangent:
$$ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} $$

Since $ \tan 8\theta \cdot \tan 2\theta = 1 $, we can infer that:
$$ 8\theta = 90^\circ - 2\theta $$

Solving for $ \theta $:
$$ 8\theta + 2\theta = 90^\circ $$
$$ 10\theta = 90^\circ $$
$$ \theta = 9^\circ $$

Now, we need to find $ \tan 5\theta $:
$$ \tan 5\theta = \tan(5 \times 9^\circ) = \tan 45^\circ $$

We know that:
$$ \tan 45^\circ = 1 $$

Therefore, the value of $ \tan 5\theta $ is 1, which corresponds to Option 2.

### Explanation for Other Options:

1. **Option 1: $ \frac{1}{\sqrt{3}} $**
   - This value does not correspond to $ \tan 45^\circ $. It is the value of $ \tan 30^\circ $, which is not applicable here.

2. **Option 3: $ \sqrt{3} $**
   - This value corresponds to $ \tan 60^\circ $, which is not the angle we are dealing with in this problem.

3. **Option 4: 0**
   - This value corresponds to $ \tan 0^\circ $, which is not relevant to the given problem as $ \tan 5\theta = \tan 45^\circ $.

Thus, the correct answer is indeed Option 2, as $ \tan 5\theta = 1 $.