If `5sectheta-3tantheta=5`, then what is the value of `5tantheta-3sectheta` ?

Options

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

Answer

To solve the problem, we start with the equation given:

1. **Given**:
\[ 5\sec\theta - 3\tan\theta = 5 \]

2. **Rearranging the equation**:
We can rearrange this equation to isolate the terms involving \(\sec\theta\) and \(\tan\theta\):
\[ 5\sec\theta = 5 + 3\tan\theta \]

3. **Dividing by 5**:
Now, divide the entire equation by 5 to express \(\sec\theta\):
\[ \sec\theta = 1 + \frac{3}{5}\tan\theta \]

4. **Substituting into the expression**:
We need to find the value of:
\[ 5\tan\theta - 3\sec\theta \]
We can substitute \(\sec\theta\) from the earlier step:
\[ 5\tan\theta - 3\sec\theta = 5\tan\theta - 3\left(1 + \frac{3}{5}\tan\theta\right) \]

5. **Expanding the expression**:
Now, expand the expression:
\[ 5\tan\theta - 3 - \frac{9}{5}\tan\theta \]

6. **Combining like terms**:
To combine the terms involving \(\tan\theta\), we first express \(5\tan\theta\) in terms of a common denominator:
\[ 5\tan\theta = \frac{25}{5}\tan\theta \]
Now, we can combine:
\[ \frac{25}{5}\tan\theta - \frac{9}{5}\tan\theta - 3 = \frac{16}{5}\tan\theta - 3 \]

7. **Final expression**:
Thus, we have:
\[ 5\tan\theta - 3\sec\theta = \frac{16}{5}\tan\theta - 3 \]

8. **Using the original condition**:
Now, we can use the original condition \(5\sec\theta - 3\tan\theta = 5\) to find a numerical value. We can also use the identity:
\[ a\sec\theta - b\tan\theta = c \implies a\tan\theta - b\sec\theta = \sqrt{a^2 + b^2 - c^2} \]
Here, \(a = 5\), \(b = 3\), and \(c = 5\):
\[ x = \sqrt{5^2 + 3^2 - 5^2} = \sqrt{9} = 3 \]

9. **Conclusion**:
Therefore, the value of \(5\tan\theta - 3\sec\theta\) is:
\[ 3 \]