If `(1)/(costheta+sectheta)=(1)/(2)` , what is the value of `cos^(100)theta+sec^(100)theta`?

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the equation \(\frac{1}{\cos \theta + \sec \theta} = \frac{1}{2}\), we can follow these steps:

### Step 1: Cross-multiply the equation
Starting with the given equation:
\[ \frac{1}{\cos \theta + \sec \theta} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 1 = \frac{1}{2}(\cos \theta + \sec \theta) \]

### Step 2: Multiply both sides by 2
Multiplying both sides by 2, we have:
\[ 2 = \cos \theta + \sec \theta \]

### Step 3: Substitute \(\sec \theta\) with \(\frac{1}{\cos \theta}\)
Recall that \(\sec \theta = \frac{1}{\cos \theta}\). Substituting this into the equation gives:
\[ 2 = \cos \theta + \frac{1}{\cos \theta} \]

### Step 4: Let \(x = \cos \theta\)
Let \(x = \cos \theta\). Then the equation becomes:
\[ 2 = x + \frac{1}{x} \]

### Step 5: Multiply through by \(x\)
To eliminate the fraction, multiply through by \(x\):
\[ 2x = x^2 + 1 \]

### Step 6: Rearrange the equation
Rearranging gives us:
\[ x^2 - 2x + 1 = 0 \]

### Step 7: Factor the quadratic
This can be factored as:
\[ (x - 1)^2 = 0 \]

### Step 8: Solve for \(x\)
Thus, we find:
\[ x - 1 = 0 \implies x = 1 \]
Since \(x = \cos \theta\), we have:
\[ \cos \theta = 1 \]

### Step 9: Find \(\sec \theta\)
Now, since \(\sec \theta = \frac{1}{\cos \theta}\):
\[ \sec \theta = \frac{1}{1} = 1 \]

### Step 10: Calculate \(\cos^{100} \theta + \sec^{100} \theta\)
Now we can calculate:
\[ \cos^{100} \theta + \sec^{100} \theta = 1^{100} + 1^{100} = 1 + 1 = 2 \]

### Final Answer
Thus, the value of \(\cos^{100} \theta + \sec^{100} \theta\) is:
\[ \boxed{2} \]