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!