Hence, #costheta = -4/5#. We know that #tantheta>0#, and the only quadrant where tangent is positive and the other trig ratios are negative is quadrant III.
Since we only know the side adjacent #theta# and the hypotenuse, we must find the side opposite #theta#.
We can do this using Pythagorean theorem. Let a be #-4# and #c# be 5.
#a^2 + b^2 = c^2#
#b^2 = c^2 - a^2#
#b^2 = (5)^2 - (-4)^2#
#b ^2= 25 - 16#
#b = sqrt(9)#
#b = +-3#
We will take the #-3#, because in quadrant three both opposite and adjacent sides to the angle #theta# will be negative.
Now that we know that
#"adjacent = -4"#
#"opposite"= -3#
#"hypotenuse = 5"#
We can define cosine and cotangent. Cosine is adjacent/hypotenuse, and cotangent is #1/tantheta = 1/("opposite"/"adjacent") = "adjacent"/"opposite"#.
Applying these definitions to the problem at hand, we have:
#cottheta = 4/3#
#costheta = -4/5#
Now, adding these is simple arithmetic.
#4/3 + (-4/5) = 20/15 - 12/15 = 8/15#
Thus, #costheta + cot theta = 8/15#.
Hopefully this helps!
