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Let $0<a<\pi/2,0<b<\pi/2$, $0<\lambda<1, \mu=1-\lambda$. Does anyone see a good proof of the inequality:

$$\sin(\lambda a)\sin(\lambda b)+\sin(\lambda a)\sin(\mu b)\cos(b)+\sin(\lambda b)\sin(\mu a)\cos(a)+\sin(\mu a)\sin(\mu b)>\sin(a)\sin(b).$$

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The LHS equals the RHS for $a=b=\pi/4,\lambda=\mu=1/2$. – mathlove Dec 19 '15 at 21:00

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