证明sina×cosa≤1/2

(sin a-cos a)^2>=0
sin ^2a+cos ^2a-2sin a* cos a>=0
1-2sin a* cos a>=0
2sin a* cos a<=1
sin a* cos a<=1/2

sin a + cos a
=√2(√2/2sin a +√2/2 cos a)
=√2(sin∏/4*sin a +cos∏/4 cos a)
=√2cos(∏/4 - a)
-1<=cos(∏/4 - a)<=1
-√2<=√2cos(∏/4 - a)<=√2

所以sin a + cos a∈[-√2，√2]

sina*cosa=1/2*2sinacnsa=1/2sin2a,sin2a<=1,所以sina*cosa<=1/2