If `costheta+sectheta=2(0^(0)lethetale90^(@))`, then `cos100theta+sin100theta=`?

Options

A. 0
B. 1
C. 2
D. 3

Answer

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

\[ \cos \theta + \sec \theta = 2 \]

### Step 1: Rewrite the equation
We know that \(\sec \theta = \frac{1}{\cos \theta}\). Therefore, we can rewrite the equation as:

\[ \cos \theta + \frac{1}{\cos \theta} = 2 \]

### Step 2: Multiply through by \(\cos \theta\)
To eliminate the fraction, we multiply both sides by \(\cos \theta\):

\[ \cos^2 \theta + 1 = 2 \cos \theta \]

### Step 3: Rearrange the equation
Rearranging gives us a standard quadratic equation:

\[ \cos^2 \theta - 2 \cos \theta + 1 = 0 \]

### Step 4: Factor the quadratic
This can be factored as:

\[ (\cos \theta - 1)^2 = 0 \]

### Step 5: Solve for \(\cos \theta\)
Setting the factor equal to zero gives:

\[ \cos \theta - 1 = 0 \implies \cos \theta = 1 \]

### Step 6: Find \(\theta\)
The value of \(\theta\) that satisfies \(\cos \theta = 1\) is:

\[ \theta = 0^\circ \]

### Step 7: Substitute \(\theta\) into the expression
Now we need to find:

\[ \cos(100\theta) + \sin(100\theta) \]

Substituting \(\theta = 0^\circ\):

\[ \cos(100 \times 0) + \sin(100 \times 0) = \cos(0) + \sin(0) \]

### Step 8: Calculate the values
We know:

\[ \cos(0) = 1 \quad \text{and} \quad \sin(0) = 0 \]

Thus:

\[ \cos(0) + \sin(0) = 1 + 0 = 1 \]

### Final Answer
Therefore, the value of \(\cos(100\theta) + \sin(100\theta)\) is:

\[ \boxed{1} \]

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