Show that if $ 0\leq x\leq\pi/2 $ $$2^{\sqrt{\sin x}}+2^{\sqrt{\cos x}}\geq 2+\sin x+\cos x \geq 3$$ I encountered this problem when I tried to solve the equation $$2^{\sqrt{\sin x}}+2^{\sqrt{\cos x}}=2+\sin x+\cos x$$ I tried to use $A.M\geq G.M$ but it didn't help or I could not solve it properly.
2026-04-13 21:15:51.1776114951
Inequality involving trigonometry and exponentiation
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Hint: for $0\leqslant x\leqslant1$ $$2^x\geqslant 1 + x^2$$