Given : `4 sin theta = 3 cos theta`, find the value of :
(i) `sin theta`
(ii) `cos theta`
(iii) `cot^(2) theta - "cosec"^(2) theta`
(iv) `4 cos^(2) theta - 3 sin^(2) theta + 2`

Answer

To solve the given problem step by step, we start with the equation provided:

**Given:**
\[ 4 \sin \theta = 3 \cos \theta \]

### Step 1: Find \(\sin \theta\) and \(\cos \theta\)

1. Rearranging the equation:
\[ \frac{\sin \theta}{\cos \theta} = \frac{3}{4} \]
This implies:
\[ \tan \theta = \frac{3}{4} \]

2. Now, we can visualize this as a right triangle where the opposite side (perpendicular) is 3 and the adjacent side (base) is 4.

3. Using the Pythagorean theorem to find the hypotenuse:
\[ \text{Hypotenuse} = \sqrt{(3^2 + 4^2)} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

4. Now we can find \(\sin \theta\) and \(\cos \theta\):
- \(\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5}\)
- \(\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5}\)

### Step 2: Find \(\cot^2 \theta - \csc^2 \theta\)

1. First, we find \(\cot \theta\):
\[ \cot \theta = \frac{1}{\tan \theta} = \frac{4}{3} \]
Therefore:
\[ \cot^2 \theta = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]

2. Next, we find \(\csc \theta\):
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{5}{3} \]
Therefore:
\[ \csc^2 \theta = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \]

3. Now we can calculate \(\cot^2 \theta - \csc^2 \theta\):
\[ \cot^2 \theta - \csc^2 \theta = \frac{16}{9} - \frac{25}{9} = \frac{16 - 25}{9} = \frac{-9}{9} = -1 \]

### Step 3: Find \(4 \cos^2 \theta - 3 \sin^2 \theta + 2\)

1. Calculate \(\cos^2 \theta\) and \(\sin^2 \theta\):
- \(\cos^2 \theta = \left(\frac{4}{5}\right)^2 = \frac{16}{25}\)
- \(\sin^2 \theta = \left(\frac{3}{5}\right)^2 = \frac{9}{25}\)

2. Substitute these values into the expression:
\[ 4 \cos^2 \theta - 3 \sin^2 \theta + 2 = 4 \left(\frac{16}{25}\right) - 3 \left(\frac{9}{25}\right) + 2 \]
\[ = \frac{64}{25} - \frac{27}{25} + 2 \]
\[ = \frac{64 - 27}{25} + 2 = \frac{37}{25} + 2 \]
\[ = \frac{37}{25} + \frac{50}{25} = \frac{87}{25} \]

### Final Answers:
(i) \(\sin \theta = \frac{3}{5}\)
(ii) \(\cos \theta = \frac{4}{5}\)
(iii) \(\cot^2 \theta - \csc^2 \theta = -1\)
(iv) \(4 \cos^2 \theta - 3 \sin^2 \theta + 2 = \frac{87}{25}\)