show that `2^(sin x)+2^(cos x)ge2^(1-(1)/sqrt(2))`

Explanation

To show that \( 2^{\sin x} + 2^{\cos x} \geq 2^{1 - \frac{1}{\sqrt{2}}} \), we can utilize the Arithmetic Mean-Geometric Mean (AM-GM) inequality. Here’s a step-by-step solution:

### Step 1: Apply the AM-GM Inequality
We start by applying the AM-GM inequality to the terms \( 2^{\sin x} \) and \( 2^{\cos x} \).

\[ \frac{2^{\sin x} + 2^{\cos x}}{2} \geq \sqrt{2^{\sin x} \cdot 2^{\cos x}} \]
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