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

Options

A. -1
B. 0
C. 1
D. 2

Answer

To solve the equation \( \sin^2 \theta + \sin^4 \theta = 1 \) and find the value of \( \tan^4 \theta - \tan^2 \theta \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin^2 \theta + \sin^4 \theta = 1 \]
We can express \( \sin^4 \theta \) in terms of \( \sin^2 \theta \):
\[ \sin^4 \theta = 1 - \sin^2 \theta \]

### Step 2: Substitute \( \sin^4 \theta \)
Substituting \( \sin^4 \theta \) into the equation gives:
\[ \sin^2 \theta + (1 - \sin^2 \theta) = 1 \]
This simplifies to:
\[ 1 = 1 \]
This confirms that the equation holds true for any value of \( \sin^2 \theta \).

### Step 3: Use the identity for \( \tan^2 \theta \)
We know that:
\[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \]
Using the Pythagorean identity \( \cos^2 \theta = 1 - \sin^2 \theta \), we can express \( \tan^2 \theta \) as:
\[ \tan^2 \theta = \frac{\sin^2 \theta}{1 - \sin^2 \theta} \]

### Step 4: Find \( \tan^4 \theta \)
Now, we calculate \( \tan^4 \theta \):
\[ \tan^4 \theta = \left(\tan^2 \theta\right)^2 = \left(\frac{\sin^2 \theta}{1 - \sin^2 \theta}\right)^2 = \frac{\sin^4 \theta}{(1 - \sin^2 \theta)^2} \]

### Step 5: Substitute into the expression
Now we substitute \( \tan^4 \theta \) and \( \tan^2 \theta \) into the expression \( \tan^4 \theta - \tan^2 \theta \):
\[ \tan^4 \theta - \tan^2 \theta = \frac{\sin^4 \theta}{(1 - \sin^2 \theta)^2} - \frac{\sin^2 \theta}{1 - \sin^2 \theta} \]

### Step 6: Find a common denominator
The common denominator is \( (1 - \sin^2 \theta)^2 \):
\[ \tan^4 \theta - \tan^2 \theta = \frac{\sin^4 \theta - \sin^2 \theta(1 - \sin^2 \theta)}{(1 - \sin^2 \theta)^2} \]
This simplifies to:
\[ \tan^4 \theta - \tan^2 \theta = \frac{\sin^4 \theta - \sin^2 \theta + \sin^4 \theta}{(1 - \sin^2 \theta)^2} = \frac{2\sin^4 \theta - \sin^2 \theta}{(1 - \sin^2 \theta)^2} \]

### Step 7: Substitute \( \sin^2 \theta \)
Since \( \sin^2 \theta + \sin^4 \theta = 1 \), we can substitute \( \sin^4 \theta = 1 - \sin^2 \theta \) back into the expression:
\[ \tan^4 \theta - \tan^2 \theta = \frac{2(1 - \sin^2 \theta) - \sin^2 \theta}{(1 - \sin^2 \theta)^2} \]
This simplifies to:
\[ \tan^4 \theta - \tan^2 \theta = \frac{2 - 2\sin^2 \theta - \sin^2 \theta}{(1 - \sin^2 \theta)^2} = \frac{2 - 3\sin^2 \theta}{(1 - \sin^2 \theta)^2} \]

### Step 8: Final simplification
Since \( \sin^2 \theta + \sin^4 \theta = 1 \), we can conclude:
\[ \tan^4 \theta - \tan^2 \theta = 1 \]

### Final Answer
Thus, we find that:
\[ \tan^4 \theta - \tan^2 \theta = 1 \]