The inequality `2^(sintheta)+2^(costheta)ge2^(1-1/sqrt(2))`,holds for all real values of `theta`

Answer

To solve the inequality \( 2^{\sin \theta} + 2^{\cos \theta} \geq 2^{1 - \frac{1}{\sqrt{2}}} \) for all real values of \( \theta \), we will follow these steps:

### Step 1: Rewrite the Inequality
We start with the given inequality:
\[ 2^{\sin \theta} + 2^{\cos \theta} \geq 2^{1 - \frac{1}{\sqrt{2}}} \]

### Step 2: Apply the Arithmetic Mean-Geometric Mean (AM-GM) Inequality
We can apply the AM-GM inequality, which states that for any non-negative numbers \( a \) and \( b \):
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
Let \( a = 2^{\sin \theta} \) and \( b = 2^{\cos \theta} \). Then we have:
\[ \frac{2^{\sin \theta} + 2^{\cos \theta}}{2} \geq \sqrt{2^{\sin \theta} \cdot 2^{\cos \theta}} \]
This simplifies to:
\[ 2^{\sin \theta} + 2^{\cos \theta} \geq 2 \sqrt{2^{\sin \theta + \cos \theta}} = 2 \cdot 2^{\frac{\sin \theta + \cos \theta}{2}} = 2^{1 + \frac{\sin \theta + \cos \theta}{2}} \]

### Step 3: Analyze \( \sin \theta + \cos \theta \)
We know that:
\[ \sin \theta + \cos \theta = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \]
This means:
\[ \sin \theta + \cos \theta \leq \sqrt{2} \]
Thus,
\[ \frac{\sin \theta + \cos \theta}{2} \leq \frac{\sqrt{2}}{2} \]

### Step 4: Substitute Back
Substituting this back into our inequality gives:
\[ 2^{1 + \frac{\sin \theta + \cos \theta}{2}} \geq 2^{1 + \frac{\sqrt{2}}{2}} \]
This means:
\[ 2^{\sin \theta} + 2^{\cos \theta} \geq 2^{1 + \frac{\sqrt{2}}{2}} \]

### Step 5: Compare with the Right Side
We need to show that:
\[ 2^{1 + \frac{\sqrt{2}}{2}} \geq 2^{1 - \frac{1}{\sqrt{2}}} \]
This simplifies to:
\[ 1 + \frac{\sqrt{2}}{2} \geq 1 - \frac{1}{\sqrt{2}} \]
Subtracting 1 from both sides gives:
\[ \frac{\sqrt{2}}{2} \geq -\frac{1}{\sqrt{2}} \]
Multiplying through by \( \sqrt{2} \) (which is positive) gives:
\[ 1 \geq -1 \]
This is always true.

### Conclusion
Thus, we have shown that:
\[ 2^{\sin \theta} + 2^{\cos \theta} \geq 2^{1 - \frac{1}{\sqrt{2}}} \]
holds for all real values of \( \theta \).