Question

·已知tan[(a+b)/2]＝(√6)/2,tanatanb=13/7,求cos(a-b)=____.

Answer 1

tan(a+b)

=2tan[(a+b)/2]/{1-{tan[(a+b)/2]}^2}

=√6/(1-6/4)

=-2√6

tana+tanb

=tan(a+b)*(1-tanatanb)

=-2√6*(1-13/7)

=(12√6)/7

(tana-tanb)^2

=(tana+tanb)^2-4tanatanb

=864/49-52/7

=500/49

tan(a-b)=(tana-tanb)/(1+tanatanb)

[tan(a-b)]^2

=(tana-tanb)^2/(1+tanatanb)^2

=(500/49)/(1+13/7)^2

=5/4

1+[tan(a-b)]^2=1/[cos(a-b)]^2

1+5/4=1/[cos(a-b)]^2

cos(a-b)=2/3

tana和tanb均为正数,tan[(a+b)/2]也为正数且大于1,所以a、b同象限,cos(a-b)为正