Consider f, g and h be three real valued function defined on R.
Let `f(x)= sin 3x + cos x , g(x)= cos 3x + sin x ` and ` h(x) = f^(2)(x) + g^(2)(x) `
Q. General solution of the equation ` h(x) = 4 ` , is :
[where ` n in I ` ]

Options

A. `(pi)/(8)`
B. `(pi)/(4)`
C. `(pi)/(6)`
D. `(pi)/(2)`

Answer

To solve the problem step by step, we will find the general solution of the equation \( h(x) = 4 \) where \( h(x) = f^2(x) + g^2(x) \).

### Step 1: Define the functions
We are given:
- \( f(x) = \sin(3x) + \cos(x) \)
- \( g(x) = \cos(3x) + \sin(x) \)

### Step 2: Write the expression for \( h(x) \)
Using the definitions of \( f(x) \) and \( g(x) \):
\[ h(x) = f^2(x) + g^2(x) = (\sin(3x) + \cos(x))^2 + (\cos(3x) + \sin(x))^2 \]

### Step 3: Expand \( h(x) \)
Now we will expand both squares:
1. For \( f^2(x) \):
\[ f^2(x) = (\sin(3x) + \cos(x))^2 = \sin^2(3x) + \cos^2(x) + 2\sin(3x)\cos(x) \]
2. For \( g^2(x) \):
\[ g^2(x) = (\cos(3x) + \sin(x))^2 = \cos^2(3x) + \sin^2(x) + 2\cos(3x)\sin(x) \]

Combining these, we have:
\[ h(x) = \sin^2(3x) + \cos^2(x) + 2\sin(3x)\cos(x) + \cos^2(3x) + \sin^2(x) + 2\cos(3x)\sin(x) \]

### Step 4: Use the Pythagorean identity
Using the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \):
\[ h(x) = 1 + 1 + 2\sin(3x)\cos(x) + 2\cos(3x)\sin(x) \]
This simplifies to:
\[ h(x) = 2 + 2(\sin(3x)\cos(x) + \cos(3x)\sin(x)) \]

### Step 5: Use the sine addition formula
Using the sine addition formula \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \):
\[ h(x) = 2 + 2\sin(4x) \]

### Step 6: Set \( h(x) = 4 \)
Now we set the equation:
\[ 2 + 2\sin(4x) = 4 \]

### Step 7: Solve for \( \sin(4x) \)
Subtract 2 from both sides:
\[ 2\sin(4x) = 2 \]
Divide by 2:
\[ \sin(4x) = 1 \]

### Step 8: Find the general solution for \( \sin(4x) = 1 \)
The sine function equals 1 at:
\[ 4x = \frac{\pi}{2} + 2n\pi \quad \text{for } n \in \mathbb{Z} \]
Dividing by 4 gives:
\[ x = \frac{\pi}{8} + \frac{n\pi}{2} \]

### Final Answer
Thus, the general solution of the equation \( h(x) = 4 \) is:
\[ x = \frac{\pi}{8} + \frac{n\pi}{2}, \quad n \in \mathbb{Z} \]