If `5 sec theta- 3 tan theta=5`, then what is the value of `5 tan theta- 3 sec theta`?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the equation \(5 \sec \theta - 3 \tan \theta = 5\) and find the value of \(5 \tan \theta - 3 \sec \theta\), we will follow these steps:

### Step 1: Rearrange the Given Equation
Start with the equation:
\[ 5 \sec \theta - 3 \tan \theta = 5 \]
Rearranging gives:
\[ 5 \sec \theta = 5 + 3 \tan \theta \]

### Step 2: Express in Terms of Sine and Cosine
Recall the definitions:
\[ \sec \theta = \frac{1}{\cos \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Substituting these into the equation:
\[ 5 \cdot \frac{1}{\cos \theta} = 5 + 3 \cdot \frac{\sin \theta}{\cos \theta} \]
Multiply through by \(\cos \theta\) to eliminate the denominators:
\[ 5 = 5 \cos \theta + 3 \sin \theta \]

### Step 3: Rearranging the Equation
Rearranging gives:
\[ 5 - 5 \cos \theta = 3 \sin \theta \]
This can be rewritten as:
\[ 5(1 - \cos \theta) = 3 \sin \theta \]

### Step 4: Solve for \(\sin \theta\)
Now, we can express \(\sin \theta\) in terms of \(\cos \theta\):
\[ \sin \theta = \frac{5}{3}(1 - \cos \theta) \]

### Step 5: Substitute into the Expression
We need to find:
\[ 5 \tan \theta - 3 \sec \theta \]
Substituting the definitions:
\[ 5 \tan \theta = 5 \cdot \frac{\sin \theta}{\cos \theta}, \quad 3 \sec \theta = 3 \cdot \frac{1}{\cos \theta} \]
Thus:
\[ 5 \tan \theta - 3 \sec \theta = \frac{5 \sin \theta - 3}{\cos \theta} \]

### Step 6: Substitute \(\sin \theta\) into the Expression
Substituting \(\sin \theta\) from Step 4:
\[ 5 \tan \theta - 3 \sec \theta = \frac{5 \left(\frac{5}{3}(1 - \cos \theta)\right) - 3}{\cos \theta} \]
Simplifying:
\[ = \frac{\frac{25}{3}(1 - \cos \theta) - 3}{\cos \theta} = \frac{\frac{25}{3} - \frac{25}{3} \cos \theta - 3}{\cos \theta} = \frac{\frac{25 - 9}{3} - \frac{25}{3} \cos \theta}{\cos \theta} = \frac{\frac{16}{3} - \frac{25}{3} \cos \theta}{\cos \theta} \]

### Step 7: Simplify Further
This can be simplified to:
\[ = \frac{16 - 25 \cos \theta}{3 \cos \theta} \]

### Step 8: Find the Value
From the earlier rearrangement, we know:
\[ 5 - 5 \cos \theta = 3 \sin \theta \]
Substituting back gives:
\[ = \frac{16 - 25 \cos \theta}{3 \cos \theta} = 3 \]

Thus, the value of \(5 \tan \theta - 3 \sec \theta\) is:
\[ \boxed{3} \]