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

Options

A. `(3)/(4)`
B. `(3)/(2)`
C. `(1)/(2)`
D. 1

Answer

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

### Step 1: Substitute \( x = \tan^2 \theta \)
Let \( x = \tan^2 \theta \). Then the equation becomes:
\[ x^2 + x - 2 = 0 \]

### Step 2: Factor the quadratic equation
We can factor the quadratic equation:
\[ (x - 1)(x + 2) = 0 \]

### Step 3: Solve for \( x \)
Setting each factor to zero gives us:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \quad (\text{not valid since } x = \tan^2 \theta \geq 0) \]
Thus, we have \( \tan^2 \theta = 1 \).

### Step 4: Find \( \sec^2 \theta \)
Using the identity \( \sec^2 \theta = 1 + \tan^2 \theta \):
\[ \sec^2 \theta = 1 + 1 = 2 \]

### Step 5: Relate \( \cos^2 \theta \) to \( \sec^2 \theta \)
Since \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \), we can find \( \cos^2 \theta \):
\[ \cos^2 \theta = \frac{1}{\sec^2 \theta} = \frac{1}{2} \]

### Step 6: Calculate \( \cos^4 \theta + \cos^2 \theta \)
Now we can find \( \cos^4 \theta + \cos^2 \theta \):
\[ \cos^4 \theta = (\cos^2 \theta)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
So,
\[ \cos^4 \theta + \cos^2 \theta = \frac{1}{4} + \frac{1}{2} \]

### Step 7: Find a common denominator and simplify
To add these fractions, we find a common denominator:
\[ \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \]

### Final Answer
Thus, the value of \( \cos^4 \theta + \cos^2 \theta \) is:
\[ \boxed{\frac{3}{4}} \]