Start with the identity:
#cos(x-y) = cos(y)cos(x)+sin(y)sin(x)" [1]"#
We are given that #cos(x) = 15/17#
#cos(x-y) = cos(y)15/17+sin(y)sin(x)" [1.1]"#
Use the identity
#sin(x) = +-sqrt(1-cos^2(x))#
We are told that #0^@ < x < 90^@#, therefore, we shall use the positive case:
#sin(x) = sqrt(1-cos^2(x))#
Substitute #cos^2(x) = (15/17)^2#
#sin(x) = sqrt(1-(15/17)^2)#
#sin(x) = sqrt(289/289-225/289)#
#sin(x) = sqrt(64/289)#
#sin(x) = 8/17" [2]"#
Substitute equation [2] into equation [1.1]:
#cos(x-y) = cos(y)15/17+sin(y)8/17" [1.2]"#
Use the identity
#1+cot^2(y) = csc^2(y)#
The must be an error because y is the second quadrant but the cotangent is positive; I shall assume that it is negative. Substitute #cot^2(y) = (-24/7)^2#
#1+ (-24/7)^2=csc^2(y)#
#49/49+ 576/49=csc^2(y)#
#625/49=csc^2(y)#
Substitute #csc^2(y) = 1/sin^2(y)#
#625/49=1/sin^2(y)#
#sin^2(y) = 49/625#
#sin(y) = +-7/25#
Because we are told that #90^@ < y < 180^@#, we shall choose the positive value:
#sin(y) = 7/25" [3]"#
Substitute equation [3] into equation [1.2]:
#cos(x-y) = cos(y)15/17+7/25(8/17)" [1.2]"#
Use the identity:
#cot(y) = cos(y)/sin(y)#
Again assuming that the cotangent is negative:
#-24/7 = cos(y)/(7/25)#
#cos(y) = -24/7 7/25#
#cos(y) = -24/25" [4]"#
Substitute equation [4] into equation [1.2]:
#cos(x-y) = -24/25 (15/17)+7/25 (8/17)#
#cos(x-y) = -304/425#
