Find the exact value of tan(a-b)

sin a = 4/5, -3pi/2<a<-pi;
tan b = -sqrt2, pi/2<b<pi

identity used is:

tan(a-b)=(tan a-tan b)/1+tan a tan b

simplify answer using radicals.

(a is alpha, b is beta)

Both the sine and cosine curves are the same for

-3pi/2<a<-pi as they are for π/2<a<π (2nd quad)
so if sina = 4/5, then cosa = -3/5
and tana = -4/3

then tan(a-b)
= (tana - tanb)/(1 + tanatanb)
= ((-4/3) - (-√2))/( 1 + (-4/3)(-√2))
= (√2 - 4/3)/(1 + 4√2/3)
= (3√2 - 4)/(3 + 4√2)

I don't know if you have to rationalize that, if you do carefully multiply top and bottom by (3 - 4√2)

I put that in but it said the answer was

36-25�ã2/23
any idea how that works?

That is exactly what my answer works out to if you rationalize it.

I had suggested to do that.

8 cot(A) − 8/

1 + tan(−A)

To find the exact value of tan(a-b), we can use the given values of sin a and tan b along with the identity tan(a-b) = (tan a - tan b) / (1 + tan a * tan b).

Given:
sin a = 4/5, where -3π/2 < a < -π
tan b = -√2, where π/2 < b < π

Step 1: Find the value of tan a.
Since sin a = 4/5, we can use the Pythagorean identity sin^2 a + cos^2 a = 1 to find cos a.
cos a = √(1 - sin^2 a) = √(1 - (4/5)^2) = √(1 - 16/25) = √(9/25) = 3/5

Using the definition of tangent, tan a = sin a / cos a = (4/5) / (3/5) = 4/3.

Step 2: Find the value of tan(a-b).
Using the given identity tan(a-b) = (tan a - tan b) / (1 + tan a * tan b), we can substitute the values:

tan(a-b) = (tan a - tan b) / (1 + tan a * tan b)
= (4/3 - (-√2)) / (1 + (4/3) * (-√2))
= (4/3 + √2) / (1 - (4/3 * √2))

Step 3: Simplify the answer using radicals.
To rationalize the denominator, we can multiply the numerator and denominator by the conjugate of the denominator, which is (1 + (4/3 * √2)).

tan(a-b) = [(4/3 + √2) / (1 - (4/3 * √2))] * [(1 + (4/3 * √2)) / (1 + (4/3 * √2))]
= [(4/3 + √2)(1 + (4/3 * √2))] / [1 - (4/3 * √2)]^2

Expanding the numerator:
= [4/3 + √2 + (16/9 * √2) + (16/9 * 2)] / [1 - (4/3 * √2)]^2
= [4/3 + √2 + (32/9 * √2) + (32/9)] / [1 - (4/3 * √2)]^2
= [(4/3 + 32/9) + (√2 + (32/9 * √2))] / [1 - (4/3 * √2)]^2
= [(40/9) + ((9 + 32)/9) * √2] / [1 - (4/3 * √2)]^2
= [(40/9) + (41/9) * √2] / [1 - (4/3 * √2)]^2

Therefore, the exact value of tan(a-b) is [(40/9) + (41/9) * √2] / [1 - (4/3 * √2)]^2.