We are given that [tex]\(\sin x = \frac{8}{17}\)[/tex] and [tex]\(\cos y = \frac{3}{5}\)[/tex]. We need to find [tex]\(\tan(x - y)\)[/tex].
First, we use the Pythagorean identity to find [tex]\(\cos x\)[/tex] and [tex]\(\sin y\)[/tex].
For [tex]\(\cos x\)[/tex]:
[tex]\[
\cos^2 x = 1 - \sin^2 x
\][/tex]
[tex]\[
\cos^2 x = 1 - \left(\frac{8}{17}\right)^2
\][/tex]
[tex]\[
\cos^2 x = 1 - \frac{64}{289}
\][/tex]
[tex]\[
\cos^2 x = \frac{289}{289} - \frac{64}{289}
\][/tex]
[tex]\[
\cos^2 x = \frac{225}{289}
\][/tex]
[tex]\[
\cos x = \sqrt{\frac{225}{289}}
\][/tex]
[tex]\[
\cos x = \frac{15}{17}
\][/tex]
For [tex]\(\sin y\)[/tex]:
[tex]\[
\sin^2 y = 1 - \cos^2 y
\][/tex]
[tex]\[
\sin^2 y = 1 - \left(\frac{3}{5}\right)^2
\][/tex]
[tex]\[
\sin^2 y = 1 - \frac{9}{25}
\][/tex]
[tex]\[
\sin^2 y = \frac{25}{25} - \frac{9}{25}
\][/tex]
[tex]\[
\sin^2 y = \frac{16}{25}
\][/tex]
[tex]\[
\sin y = \sqrt{\frac{16}{25}}
\][/tex]
[tex]\[
\sin y = \frac{4}{5}
\][/tex]
Next, we use the values of [tex]\(\sin x\)[/tex], [tex]\(\cos x\)[/tex], [tex]\(\sin y\)[/tex], and [tex]\(\cos y\)[/tex] to find [tex]\(\tan x\)[/tex] and [tex]\(\tan y\)[/tex].
[tex]\[
\tan x = \frac{\sin x}{\cos x}
\][/tex]
[tex]\[
\tan x = \frac{\frac{8}{17}}{\frac{15}{17}} = \frac{8}{15}
\][/tex]
[tex]\[
\tan y = \frac{\sin y}{\cos y}
\][/tex]
[tex]\[
\tan y = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}
\][/tex]
Using the tangent subtraction formula:
[tex]\[
\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}
\][/tex]
[tex]\[
\tan(x - y) = \frac{\frac{8}{15} - \frac{4}{3}}{1 + \frac{8}{15} \cdot \frac{4}{3}}
\][/tex]
[tex]\[
\tan(x - y) = \frac{\frac{8}{15} - \frac{20}{15}}{1 + \frac{32}{45}}
\][/tex]
[tex]\[
\tan(x - y) = \frac{\frac{8 - 20}{15}}{1 + \frac{32}{45}}
\][/tex]
[tex]\[
\tan(x - y) = \frac{-\frac{12}{15}}{1 + \frac{32}{45}}
\][/tex]
[tex]\[
\tan(x - y) = \frac{-\frac{4}{5}}{\frac{45 + 32}{45}}
\][/tex]
[tex]\[
\tan(x - y) = \frac{-\frac{4}{5}}{\frac{77}{45}}
\][/tex]
[tex]\[
\tan(x - y) = -\frac{4}{5} \cdot \frac{45}{77}
\][/tex]
[tex]\[
\tan(x - y) = -\frac{180}{385}
\][/tex]
[tex]\[
\tan(x - y) = -\frac{36}{77}
\][/tex]
So, the exact value of [tex]\(\tan(x - y)\)[/tex] is [tex]\(-\frac{36}{77}\)[/tex], which corresponds to answer choice:
b. [tex]\(-\frac{36}{77}\)[/tex]
