If : `sin^(2)theta+sin^(4)theta=1, "then": cot^(2)theta+cot^(4)theta=`

Options

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

Answer

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

### Step 1: Start with the given equation
We have:
\[ \sin^2 \theta + \sin^4 \theta = 1 \]

### Step 2: Rewrite \(\sin^4 \theta\)
We can express \(\sin^4 \theta\) in terms of \(\sin^2 \theta\):
\[ \sin^4 \theta = (\sin^2 \theta)^2 \]
So, we can rewrite the equation as:
\[ \sin^2 \theta + (\sin^2 \theta)^2 = 1 \]

### Step 3: Let \(x = \sin^2 \theta\)
Let \(x = \sin^2 \theta\). Then the equation becomes:
\[ x + x^2 = 1 \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ x^2 + x - 1 = 0 \]

### Step 5: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 1\), and \(c = -1\):
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 + 4}}{2} \]
\[ x = \frac{-1 \pm \sqrt{5}}{2} \]

### Step 6: Determine \(\sin^2 \theta\)
Since \(x = \sin^2 \theta\) must be non-negative, we take the positive root:
\[ \sin^2 \theta = \frac{-1 + \sqrt{5}}{2} \]

### Step 7: Find \(\cos^2 \theta\)
Using the Pythagorean identity:
\[ \cos^2 \theta = 1 - \sin^2 \theta \]
Substituting the value of \(\sin^2 \theta\):
\[ \cos^2 \theta = 1 - \frac{-1 + \sqrt{5}}{2} = \frac{2 + 1 - \sqrt{5}}{2} = \frac{3 - \sqrt{5}}{2} \]

### Step 8: Calculate \(\cot^2 \theta\)
We know that:
\[ \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \]
Substituting the values we found:
\[ \cot^2 \theta = \frac{\frac{3 - \sqrt{5}}{2}}{\frac{-1 + \sqrt{5}}{2}} = \frac{3 - \sqrt{5}}{-1 + \sqrt{5}} \]

### Step 9: Simplify \(\cot^2 \theta\)
To simplify:
\[ \cot^2 \theta = \frac{(3 - \sqrt{5})(-1 - \sqrt{5})}{(-1 + \sqrt{5})(-1 - \sqrt{5})} \]
Calculating the denominator:
\[ (-1 + \sqrt{5})(-1 - \sqrt{5}) = 1 - 5 = -4 \]
Calculating the numerator:
\[ (3 - \sqrt{5})(-1 - \sqrt{5}) = -3 - 3\sqrt{5} + \sqrt{5} + 5 = 2 - 2\sqrt{5} \]
Thus:
\[ \cot^2 \theta = \frac{2 - 2\sqrt{5}}{-4} = \frac{1 - \sqrt{5}}{-2} = \frac{\sqrt{5} - 1}{2} \]

### Step 10: Find \(\cot^4 \theta\)
Now, we need to find \(\cot^4 \theta\):
\[ \cot^4 \theta = \left(\cot^2 \theta\right)^2 = \left(\frac{\sqrt{5} - 1}{2}\right)^2 = \frac{(\sqrt{5} - 1)^2}{4} = \frac{5 - 2\sqrt{5} + 1}{4} = \frac{6 - 2\sqrt{5}}{4} = \frac{3 - \sqrt{5}}{2} \]

### Step 11: Calculate \(\cot^2 \theta + \cot^4 \theta\)
Finally, we can find:
\[ \cot^2 \theta + \cot^4 \theta = \frac{\sqrt{5} - 1}{2} + \frac{3 - \sqrt{5}}{2} = \frac{(\sqrt{5} - 1) + (3 - \sqrt{5})}{2} = \frac{2}{2} = 1 \]

### Final Answer
Thus, the required value of \(\cot^2 \theta + \cot^4 \theta\) is:
\[ \boxed{1} \]