UsingĀ $$A.M.\ge\ G.M.$$
$$\frac{\left(4^{\sin x}+4^{\cos x}\right)}{2}\ge\ \sqrt{\ 4^{\sin x+\cos x}}$$
For minimum valueĀ $$\left(4^{\sin x}+4^{\cos x}\right)=2\ \sqrt{\ 4^{\sin x+\cos x}}$$.
The minimum value of sinx +cosx when x lies in RĀ isĀ $$-\sqrt{\ 2}$$
Thus, the minimum value of the expression isĀ $$2^{1-\sqrt{\ 2}}$$
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