In ` Delta ABC , angle C = 90 ^(@) , AB = 20 and BC = 12 . D ` is a point in side AC such that CD = 9 Taking angle BDC = x, find
` tan x- cos x + 3 sin x .`

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript:

### Step 1: Draw Triangle ABC
We start by sketching triangle ABC where:
- Angle C = 90°
- AB = 20
- BC = 12

### Step 2: Find AC using Pythagorean Theorem
Using the Pythagorean theorem in triangle ABC:
\[ AB^2 = AC^2 + BC^2 \]
\[ 20^2 = AC^2 + 12^2 \]
\[ 400 = AC^2 + 144 \]
\[ AC^2 = 400 - 144 \]
\[ AC^2 = 256 \]
\[ AC = \sqrt{256} = 16 \]

### Step 3: Identify Point D on AC
Point D is on side AC such that CD = 9. Therefore, we can find AD:
\[ AD = AC - CD = 16 - 9 = 7 \]

### Step 4: Find BD using Pythagorean Theorem in Triangle BCD
Now, we apply the Pythagorean theorem in triangle BCD:
\[ BC^2 + CD^2 = BD^2 \]
\[ 12^2 + 9^2 = BD^2 \]
\[ 144 + 81 = BD^2 \]
\[ BD^2 = 225 \]
\[ BD = \sqrt{225} = 15 \]

### Step 5: Calculate Trigonometric Ratios
Now we can find the trigonometric ratios for angle \( x \) (angle BDC):
1. **tan x**:
\[ \tan x = \frac{BC}{CD} = \frac{12}{9} = \frac{4}{3} \]

2. **cos x**:
\[ \cos x = \frac{CD}{BD} = \frac{9}{15} = \frac{3}{5} \]

3. **sin x**:
\[ \sin x = \frac{BC}{BD} = \frac{12}{15} = \frac{4}{5} \]

### Step 6: Substitute into the Expression
Now we substitute these values into the expression \( \tan x - \cos x + 3 \sin x \):
\[ \tan x - \cos x + 3 \sin x = \frac{4}{3} - \frac{3}{5} + 3 \cdot \frac{4}{5} \]
Calculating \( 3 \cdot \frac{4}{5} = \frac{12}{5} \):
\[ = \frac{4}{3} - \frac{3}{5} + \frac{12}{5} \]
Combine the terms:
\[ = \frac{4}{3} + \left(\frac{12}{5} - \frac{3}{5}\right) = \frac{4}{3} + \frac{9}{5} \]

### Step 7: Find a Common Denominator
To combine \( \frac{4}{3} \) and \( \frac{9}{5} \), we find a common denominator:
- The LCM of 3 and 5 is 15.

Convert each fraction:
\[ \frac{4}{3} = \frac{4 \times 5}{3 \times 5} = \frac{20}{15} \]
\[ \frac{9}{5} = \frac{9 \times 3}{5 \times 3} = \frac{27}{15} \]

### Step 8: Combine the Fractions
Now we can combine:
\[ \frac{20}{15} + \frac{27}{15} = \frac{47}{15} \]

### Final Answer
Thus, the value of \( \tan x - \cos x + 3 \sin x \) is:
\[ \frac{47}{15} \]