If `(sin^(4)theta)/(a)+(cos^(4)theta)/(b)=(1)/(a+b),` then which one of the following is incorrect?

Options

A. `(sin^(4)theta)/(a^(2))=(cos^(4)theta)/(b^(2))`
B. `(sin^(4)theta)/(b^(2))=(cos^(4)theta)/(a^(2))`
C. `(sin^(8)theta)/(a^(3))+(cos^(8)theta)/(b^(3))=(1)/((a+b)^(3))`
D. `sin^(4)theta=(a^(2))/((a+b)^(2))`

Explanation

To solve the equation given in the question, we start with:

\[ \frac{\sin^4 \theta}{a} + \frac{\cos^4 \theta}{b} = \frac{1}{a + b} \]

### Step 1: Rewrite the equation
We can rewrite the equation as:

\[ \sin^4 \theta \cdot (a + b) + \cos^4 \theta \cdot (a + b) = \frac{(a + b)}{ab} \]

### Step 2: Multiply through by \(ab(a + b)\)
To eliminate the denominators, we multiply through by \(ab(a + b)\):

\[ \sin^4 \theta \cdot b(a + b) + \cos^4 \theta \cdot a(a + b) = 1 \]

### Step 3: Substitute \(\cos^2 \theta\)
Using the identity \(\cos^2 \theta = 1 - \sin^2 \theta\), we can express \(\cos^4 \theta\) as:

\[ \cos^4 \theta = (1 - \sin^2 \theta)^2 = 1 - 2\sin^2 \theta + \sin^4 \theta \]

### Step 4: Substitute back into the equation
Substituting this into our equation gives:

\[ \sin^4 \theta \cdot b(a + b) + (1 - 2\sin^2 \theta + \sin^4 \theta) \cdot a(a + b) = 1 \]

### Step 5: Expand and simplify
Expanding the equation results in:

\[ \sin^4 \theta \cdot b(a + b) + a(a + b) - 2a(a + b)\sin^2 \theta + a(a + b)\sin^4 \theta = 1 \]

Combining like terms gives:

\[ (b + a)\sin^4 \theta + (-2a)\sin^2 \theta + a(a + b) - 1 = 0 \]

### Step 6: Rearranging
This can be rearranged into a quadratic form in terms of \(\sin^2 \theta\):

\[ (b + a)\sin^4 \theta - 2a\sin^2 \theta + (a(a + b) - 1) = 0 \]

### Step 7: Solve for \(\sin^2 \theta\)
Let \(x = \sin^2 \theta\). The equation becomes:

\[ (b + a)x^2 - 2ax + (a(a + b) - 1) = 0 \]

Using the quadratic formula \(x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\), we can find the values of \(x\).

### Step 8: Analyze the options
Now, we need to check the options given in the question to find which one is incorrect based on the derived equations.

1. **Option 1**: \(\frac{\sin^4 \theta}{a^2} = \frac{\cos^4 \theta}{b^2}\) - Check if this holds true.
2. **Option 2**: Check the validity of this option.
3. **Option 3**: Check if \(\frac{\sin^8 \theta}{a^3} + \frac{\cos^8 \theta}{b^3} = \frac{1}{(a + b)^3}\) holds true.
4. **Option 4**: Check if \(\frac{\sin^4 \theta}{a^2} = \frac{1}{(a + b)^2}\) holds true.

After evaluating all options, we find that **Option 2** is incorrect.