If `sec^(2)theta+tan^(2)theta=sqrt(3)` , then the value of `(sec^(4)theta-tan^(4)theta)` is

Options

A. `(1)/(sqrt(3))`
B. 1
C. `sqrt(3)`
D. 0

Answer

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

\[ \sec^2 \theta + \tan^2 \theta = \sqrt{3} \]

### Step 1: Use the identity relating secant and tangent
We know that:

\[ \sec^2 \theta = 1 + \tan^2 \theta \]

Substituting this into the given equation:

\[ (1 + \tan^2 \theta) + \tan^2 \theta = \sqrt{3} \]

This simplifies to:

\[ 1 + 2\tan^2 \theta = \sqrt{3} \]

### Step 2: Isolate \(\tan^2 \theta\)
Now, we can isolate \(\tan^2 \theta\):

\[ 2\tan^2 \theta = \sqrt{3} - 1 \]

Dividing both sides by 2 gives:

\[ \tan^2 \theta = \frac{\sqrt{3} - 1}{2} \]

### Step 3: Find \(\sec^2 \theta\)
Using the identity \(\sec^2 \theta = 1 + \tan^2 \theta\):

\[ \sec^2 \theta = 1 + \frac{\sqrt{3} - 1}{2} \]

To simplify this, convert 1 into a fraction with a denominator of 2:

\[ \sec^2 \theta = \frac{2}{2} + \frac{\sqrt{3} - 1}{2} = \frac{2 + \sqrt{3} - 1}{2} = \frac{1 + \sqrt{3}}{2} \]

### Step 4: Calculate \(\sec^4 \theta - \tan^4 \theta\)
We can use the difference of squares:

\[ \sec^4 \theta - \tan^4 \theta = (\sec^2 \theta + \tan^2 \theta)(\sec^2 \theta - \tan^2 \theta) \]

We already know \(\sec^2 \theta + \tan^2 \theta = \sqrt{3}\). Now we need to find \(\sec^2 \theta - \tan^2 \theta\):

Using the identity:

\[ \sec^2 \theta - \tan^2 \theta = 1 \]

### Step 5: Substitute values
Now substituting back into the equation:

\[ \sec^4 \theta - \tan^4 \theta = (\sqrt{3})(1) = \sqrt{3} \]

### Final Answer
Thus, the value of \(\sec^4 \theta - \tan^4 \theta\) is:

\[ \sqrt{3} \]