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

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

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To prove the inequality \( 2^{\sin x} + 2^{\cos x} \geq 2^{1 - \frac{1}{\sqrt{2}}} \), we can follow these steps: ### Step 1: Apply the Arithmetic Mean-Geometric Mean Inequality (AM-GM) We start by applying the AM-GM inequality to the two positive quantities \( 2^{\sin x} \) 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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