Question

计算sin65+sin15sin10/sin25-cos15cos80的值为？

Answer 1

sin65+sin15sin10
=cos(90-65)+sin15sin10
=cos25+sin15sin10
=cos(15+10)+sin15sin10
=cos15cos10-sin15sin10+sin15sin10
=cos15cos10

sin25-cos15cos80
=cos(90-25)-cos15cos80
=cos65-cos15cos80
=cos(80-15)-cos15cos80
=cos80cos15+sin80sin15-cos15cos80
=sin80sin15

cos10=sin(90-10)=sin80

原式=cos15cos10/sin80sin15
=cos15/sin15
=cot15

附：
tan30=tan(2×15)=2tan15/(1-tan²15)=√3/3
(√3/3)tan²15+2tan15-√3/3=0
△=2²-4×(√3/3)×(-√3/3)=16/3
√△=4√3/3
tan15=(-2±4√3/3)/(2√3/3)=-√3±2
∵tan15>0
∴tan15=2-√3
则cot15=1/(2-√3)=2+√3

Answer 2

自己算