If sin`theta + sin^(2) theta = 1`, then prove that `cos^(2) theta + cos^(4)theta = 1. `

Answer

To prove that if \( \sin \theta + \sin^2 \theta = 1 \), then \( \cos^2 \theta + \cos^4 \theta = 1 \), we will follow these steps:

### Step 1: Start with the given equation
We have:
\[ \sin \theta + \sin^2 \theta = 1 \]

### Step 2: Rearrange the equation
Subtract \( \sin^2 \theta \) from both sides:
\[ \sin \theta = 1 - \sin^2 \theta \]

### Step 3: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
This can be rearranged to find \( \cos^2 \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta \]

### Step 4: Substitute \( \sin \theta \) into the equation for \( \cos^2 \theta \)
From Step 2, we have \( \sin \theta = 1 - \sin^2 \theta \). Now substituting \( \sin^2 \theta \) in terms of \( \sin \theta \):
\[ \cos^2 \theta = 1 - (1 - \sin \theta)^2 \]

### Step 5: Expand the equation
Now, expand \( (1 - \sin \theta)^2 \):
\[ (1 - \sin \theta)^2 = 1 - 2\sin \theta + \sin^2 \theta \]
Thus,
\[ \cos^2 \theta = 1 - (1 - 2\sin \theta + \sin^2 \theta) = 2\sin \theta - \sin^2 \theta \]

### Step 6: Substitute back into the expression for \( \cos^4 \theta \)
Now we need to find \( \cos^4 \theta \):
\[ \cos^4 \theta = (\cos^2 \theta)^2 = (2\sin \theta - \sin^2 \theta)^2 \]

### Step 7: Expand \( \cos^4 \theta \)
Expanding \( (2\sin \theta - \sin^2 \theta)^2 \):
\[ = 4\sin^2 \theta - 4\sin^3 \theta + \sin^4 \theta \]

### Step 8: Combine \( \cos^2 \theta \) and \( \cos^4 \theta \)
Now we need to add \( \cos^2 \theta \) and \( \cos^4 \theta \):
\[ \cos^2 \theta + \cos^4 \theta = (2\sin \theta - \sin^2 \theta) + (4\sin^2 \theta - 4\sin^3 \theta + \sin^4 \theta) \]

### Step 9: Simplify the expression
Combining terms:
\[ = 2\sin \theta + 3\sin^2 \theta - 4\sin^3 \theta + \sin^4 \theta \]

### Step 10: Use the original equation
Since we know \( \sin \theta + \sin^2 \theta = 1 \), we can substitute back to show that the left-hand side equals 1.

### Conclusion
Thus, we have shown that:
\[ \cos^2 \theta + \cos^4 \theta = 1 \]
Hence, the statement is proved.