Math
posted by Sophie on .
If α and β are two angles in Quadrant II such that tan α= 1/2 and tan β = 2/3, find cos(α+β)
Work:
cos(α+β) = [ 1  (tan α)(tan β) ] / [ 1 + (tan α)(tan β)]
cos(α+β) = [ 1  (1/2)(2/3) ] / [ 1 + (1/2)(2/3)]
cos(α+β) = [ 1  1/3 ] / [ 1 + 1/3 ]
cos(α+β) = [ 2/3 ] / [ 4/3 ]
cos(α+β) = 1/2
is this right?

Unfortunately it doesn't look right, ad... from the very start.
Could you rechceck your source from where you got the expression?
[ 1  (tan α)(tan β) ] / [ 1 + (tan α)(tan β)]
represents, according to me,
cos(α+β)/cos(αβ).
All is not lost, though.
From
1/cos^{2}(x)
=sec^{2}(x)
=1+tan^{2}(x)
tan(A+B)=(tan(A)+tan(B))/(1tan(A)tan(B))
You have ample ammunition to get the required answer.
Post your results for a checkup if you wish, or you can substitute arbitrary numbers for tan(α) and tan(β) to check by yourself. 
ohh okay thanks!

tan^1(1/2)=x=.463648
tan^1(2/3)=y.588003
x+y+pi/2 = z=2.62245
z=2.62245
cos (z) = .868243
and
tan a= 1/2
tan b= 2/3
tan (a+b)=(tan a +tan b)/(1tan a.tan b)
so tan (a+b)= 7/4
sec^2 xtan^2 x=1
so
sec (a+b)=√(1+49/16)
=√(65/16)
(a+b)=1/sec (a+b)
so cos(a+b) is positive
so cos(a+b)=.48˜.5
??? 
tan^1(1/2)=x=.463648
α=tan_{1}(1/2)=πx=2.67795
tan^1(2/3)=y.588003
β=tan_{1}(2/3)=πy=2.55359
α+β = 2.67795+2.55359 = 5.23154 rad. (serves as a check)
cos(α+β) = 0.49614 (as a check)
and
tan a= 1/2
tan b= 2/3
tan (a+b)=(tan a +tan b)/(1tan a.tan b)
so tan (a+b)= 7/4
sec^2 xtan^2 x=1
so
sec (a+b)=√(1+49/16)
=√(65/16)
Therefore
cos(α+β)
=√(16/65)
=4/√65
since α+β is in the 4th quadrant, cos(α+β)>0
Therefore
cos(α+β) = +4/√65 
oh! thanks for your time!