Evaluate tan(a-b). No decimals. Only exact answer.

Sin a = -7/25, cot b = -8/15 neither a nor b are in quadrant IV.

To evaluate tan(a-b) without using decimals and obtaining an exact answer, we can use trigonometric identities to find the required values. We'll start by finding the values of cos a and sin b, since these will be necessary to calculate tan(a-b).

Given:
sin a = -7/25,
cot b = -8/15.

We can use the Pythagorean identity to find the value of cos a:
cos^2 a = 1 - sin^2 a
cos^2 a = 1 - (-7/25)^2
cos^2 a = 1 - 49/625
cos^2 a = (625 - 49)/625
cos^2 a = 576/625
cos a = ± √(576/625)
cos a = ± 24/25

We know that a is not in quadrant IV, which means cosine is positive. Hence, we take the positive value:
cos a = 24/25

Next, we can find sin b using the identity:
cot b = cos b / sin b

Since cot b = -8/15:
-8/15 = cos b / sin b

Multiplying through by sin b:
-8 = cos b

Since cos b is negative and b is not in quadrant IV, it must be in quadrant II where sine is positive. Therefore, we can rewrite the equation as:
8/15 = cos b / sin b

Using the Pythagorean identity, we can calculate sin b:
sin^2 b = 1 - cos^2 b
sin^2 b = 1 - (8/15)^2
sin^2 b = 1 - 64/225
sin^2 b = (225 - 64)/225
sin^2 b = 161/225
sin b = ± √(161/225)
sin b = ± √(23/25)
sin b = ± √23/5

Again, since b is not in quadrant IV, sine is positive, so we take the positive value:
sin b = √23/5

Now we have all the required values, and we can proceed to calculate tan(a-b).

Using the identity:
tan(a-b) = (tan a - tan b) / (1 + tan a * tan b)

We know:
tan a = sin a / cos a = (-7/25) / (24/25) = -7/24
tan b = sin b / cos b = (√23/5) / (8/15) = (3√23)/8

Now substituting these values into tan(a-b):
tan(a-b) = (tan a - tan b) / (1 + tan a * tan b)
= (-7/24 - (3√23)/8) / (1 + ((-7/24) * ((3√23)/8)))

To simplify this further, we can multiply the numerator and denominator by 24 to remove the fractions:

tan(a-b) = (24 * (-7/24) - 24 * (3√23)/8) / (24 * 1 + ((-7/24) * (3√23)/8))
= (-7 - 9√23) / (24 + (-(21√23))/8)
= (-7 - 9√23) / (24 - (21√23)/8)

Finally, we can rationalize the denominator by multiplying both the numerator and denominator by 8:

tan(a-b) = (8*(-7) - 9√23*8) / (8*24 - (21√23))
= (-56 - 72√23) / (192 - 21√23)

Therefore, the exact value of tan(a-b) is (-56 - 72√23) / (192 - 21√23).