证明1.sin(a+b)sin(a-b)=cos^2(b)-cos^2(a)2.cosX+cosY=2cos((x+y)/2)cos((x+y)/2)拜谢@此人是山炮：

Question

证明 1.sin(a+b)sin(a-b)=cos^2(b)-cos^2(a)
2.cosX+cosY=2cos((x+y)/2)cos((x+y)/2)
拜谢
@此人是山炮：

Answer 1

sin⁡(a + b) sin⁡(a - b)
=(sin⁡a cos⁡b + cos⁡a sin⁡b )(sin⁡a cos⁡b - cos⁡a sin⁡b )
=sin^2⁡a cos^2⁡b - cos^2⁡a sin^2⁡b
=(1 - cos^2⁡a ) cos^2⁡b - (1 - cos^2⁡b ) cos^2⁡a
=(cos^2⁡b - cos^2⁡a cos^2⁡b ) - (cos^2⁡a - cos^2⁡a cos^2⁡b )
=cos^2⁡b - cos^2⁡a cos^2⁡b - cos^2⁡a + cos^2⁡a cos^2⁡b
=cos^2⁡b - cos^2⁡a
cos⁡x + cos⁡y
=cos⁡〖(x + x)/2〗 + cos⁡〖(y + y)/2〗
=cos⁡〖((x + y) + (x - y))/2〗 + cos⁡〖((x + y) - (x - y))/2〗
=cos⁡〖(x + y)/2〗 cos⁡〖(x - y)/2〗 - sin⁡〖(x + y)/2〗 sin⁡〖(x - y)/2〗 + cos⁡〖(x + y)/2〗 cos⁡〖(x - y)/2〗 + sin⁡〖(x + y)/2〗 sin⁡〖(x - y)/2〗
=cos⁡〖(x + y)/2〗 cos⁡〖(x - y)/2〗 + cos⁡〖(x + y)/2〗 cos⁡〖(x - y)/2〗
=2cos⁡〖(x + y)/2〗 cos⁡〖(x - y)/2〗