Let `theta = tan^(-1) ( tan . (5pi)/4) " and " phi = tan^(-1) ( - tan . (2pi)/3) `then

Options

A. `theta gt phi`
B. `4 theta - 3 phi = 0`
C. `theta + phi = (7pi)/12`
D. None of these

Answer

To solve the given problem, we need to evaluate the expressions for \( \theta \) and \( \phi \) and then analyze their relationships.

### Step 1: Evaluate \( \theta \)
Given:
\[ \theta = \tan^{-1}(\tan(5\pi/4)) \]

We know that:
\[ \tan(5\pi/4) = \tan(\pi + \pi/4) = \tan(\pi/4) = 1 \]
Thus:
\[ \theta = \tan^{-1}(1) \]

Since \( \tan^{-1}(1) = \frac{\pi}{4} \), we have:
\[ \theta = \frac{\pi}{4} \]

### Step 2: Evaluate \( \phi \)
Given:
\[ \phi = \tan^{-1}(-\tan(2\pi/3)) \]

We can rewrite \( 2\pi/3 \) as:
\[ 2\pi/3 = \pi - \pi/3 \]
Thus:
\[ \tan(2\pi/3) = -\tan(\pi/3) = -\sqrt{3} \]
So:
\[ \phi = \tan^{-1}(-(-\sqrt{3})) = \tan^{-1}(\sqrt{3}) \]

Since \( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \), we have:
\[ \phi = \frac{\pi}{3} \]

### Step 3: Analyze the relationship between \( \theta \) and \( \phi \)
Now we have:
\[ \theta = \frac{\pi}{4} \quad \text{and} \quad \phi = \frac{\pi}{3} \]

### Step 4: Check the equation \( 4\theta = 3\phi \)
Calculating \( 4\theta \):
\[ 4\theta = 4 \times \frac{\pi}{4} = \pi \]

Calculating \( 3\phi \):
\[ 3\phi = 3 \times \frac{\pi}{3} = \pi \]

Since \( 4\theta = 3\phi \), we can conclude:
\[ 4\theta - 3\phi = 0 \]

### Final Result
The relationship between \( \theta \) and \( \phi \) is established, and we have:
\[ 4\theta = 3\phi \]