The expression `2^(sintheta)+2^(-costheta)` is minimum when `theta` is equal to

Options

A. `2npi+(pi)/(4)`
B. `2npi+(7pi)/(4)`
C. `n pipm(pi)/(4)`
D. none of these

Explanation

To find the value of \( \theta \) for which the expression \( 2^{\sin \theta} + 2^{-\cos \theta} \) is minimized, we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Identify the Expression**:
We have the expression \( E = 2^{\sin \theta} + 2^{-\cos \theta} \).

2. **Apply AM-GM Inequality**:
According to the AM-GM inequality, for any non-negative numbers \( a \) and \( b \):
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
Here, let \( a = 2^{\sin \theta} \) and \( b = 2^{-\cos \theta} \). Thus, we have:
\[ \frac{2^{\sin \theta} + 2^{-\cos \theta}}{2} \geq \sqrt{2^{\sin \theta} \cdot 2^{-\cos \theta}} \]
This simplifies to:
\[ 2^{\sin \theta} + 2^{-\cos \theta} \geq 2 \sqrt{2^{\sin \theta - \cos \theta}} \]

3. **Find the Condition for Equality**:
The equality in AM-GM holds when \( a = b \). Therefore, we set:
\[ 2^{\sin \theta} = 2^{-\cos \theta} \]
This implies:
\[ \sin \theta = -\cos \theta \]

4. **Solve for \( \theta \)**:
The equation \( \sin \theta = -\cos \theta \) can be rewritten as:
\[ \sin \theta + \cos \theta = 0 \]
This can be solved by dividing through by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ \tan \theta = -1 \]
The general solutions for \( \tan \theta = -1 \) are:
\[ \theta = \frac{3\pi}{4} + n\pi, \quad n \in \mathbb{Z} \]

5. **Find Specific Values**:
To find specific values of \( \theta \), we can take \( n = 0 \) and \( n = 1 \):
- For \( n = 0 \): \( \theta = \frac{3\pi}{4} \)
- For \( n = 1 \): \( \theta = \frac{7\pi}{4} \)

6. **Conclusion**:
The expression \( 2^{\sin \theta} + 2^{-\cos \theta} \) is minimized at \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{7\pi}{4} \).