Find period of following functions (if existsI)
`f(x)=sin3x+tan7x`

Answer

To find the period of the function \( f(x) = \sin(3x) + \tan(7x) \), we will determine the periods of the individual components \( \sin(3x) \) and \( \tan(7x) \), and then find the least common multiple (LCM) of these periods.

### Step-by-Step Solution:

1. **Identify the Period of \( \sin(3x) \)**:
- The standard period of \( \sin(x) \) is \( 2\pi \).
- For \( \sin(kx) \), the period is given by \( \frac{2\pi}{k} \).
- Here, \( k = 3 \), so the period \( T_1 \) of \( \sin(3x) \) is:
\[ T_1 = \frac{2\pi}{3} \]

2. **Identify the Period of \( \tan(7x) \)**:
- The standard period of \( \tan(x) \) is \( \pi \).
- For \( \tan(kx) \), the period is given by \( \frac{\pi}{k} \).
- Here, \( k = 7 \), so the period \( T_2 \) of \( \tan(7x) \) is:
\[ T_2 = \frac{\pi}{7} \]

3. **Find the Least Common Multiple (LCM)**:
- We need to find the LCM of the two periods \( T_1 = \frac{2\pi}{3} \) and \( T_2 = \frac{\pi}{7} \).
- The LCM of two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) is given by:
\[ \text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{GCD}(b, d)} \]
- Here, \( a = 2\pi \), \( b = 3 \), \( c = \pi \), and \( d = 7 \).
- Calculate \( \text{LCM}(2\pi, \pi) \):
\[ \text{LCM}(2\pi, \pi) = 2\pi \]
- Calculate \( \text{GCD}(3, 7) \):
\[ \text{GCD}(3, 7) = 1 \]
- Therefore, the LCM of the periods is:
\[ T = \frac{2\pi}{1} = 2\pi \]

4. **Conclusion**:
- The period of the function \( f(x) = \sin(3x) + \tan(7x) \) is \( 2\pi \).

### Final Answer:
The period of the function \( f(x) = \sin(3x) + \tan(7x) \) is \( 2\pi \).