Use the given information to evaluate cos (a+b). Sin a = -7/25, cot b = -8/15. Neither a nor b are in quadrant IV. exact answers only. No decimals.

since the sine is negative in III or IV and the cotangent is negative in II or IV, and we are told that neither is in IV

a must be in III and b must be in II

sin a = -7/25, a in III
cos a = -24/25

cot b = -8/15 , b in II
tan b = -15/8
sinb = 15/17
cosb = -8/17

cos(a+b)
= cosa cosb - sina sinb
= (-24/25)(-8/17) - (-7/25)(15/17)
= 297/425

To evaluate cos (a+b), we can use the following trigonometric identity:

cos (a+b) = cos a * cos b - sin a * sin b

We are given the values of sin a (-7/25) and cot b (-8/15). Since cot b is the reciprocal of tan b, we can find the value of tan b using the equation:

cot b = -8/15
=> tan b = -15/8

Now, we need to find the values of cos a and cos b.

Since sin a = -7/25, we can find cos a using the Pythagorean identity:

sin^2 a + cos^2 a = 1
(-7/25)^2 + cos^2 a = 1
49/625 + cos^2 a = 1
cos^2 a = 1 - 49/625
cos^2 a = 576/625

Taking the square root of both sides, we get:

cos a = ±√(576/625)
cos a = ±(24/25)

Since a is not in quadrant IV, we know that the cosine of a is positive. Therefore:

cos a = 24/25

Now, we can substitute the values of sin a (-7/25), cos a (24/25), and tan b (-15/8) into the formula for cos (a+b):

cos (a+b) = cos a * cos b - sin a * sin b
= (24/25) * cos b - (-7/25) * (-15/8)

Since we don't have the value of cos b, we cannot simplify the expression any further without additional information.