Answer:
[tex]\frac{7}{24}[/tex]
Step-by-step explanation:
By the Pythagorean Theorem, we know
[tex]\cos^2a=1-\sin^2a[/tex].
With this, we can find [tex]\cos a[/tex] by plugging in what we know for [tex]\sin a[/tex]:
[tex]\cos^2a=1-(\frac{4}{5})^2\\~~~~~~~~=1-\frac{16}{25}\\~~~~~~~~=\frac{9}{25}[/tex].
Taking the square root, we get
[tex]\cos a=\frac{3}{5}[/tex] .
Note: there seems to be a problem with the question? It doesn't specify what range [tex]a[/tex] lies in, so we don't know whether [tex]\cos a[/tex] is positive or negative. In this case, I assumed it was positive.
From this, we can find [tex]\tan a[/tex]:
[tex]\tan a=\frac{\sin a}{\cos a}\\~~~~~~~=\dfrac{\frac{4}{5}}{\frac{3}{5}},[/tex]
so [tex]\tan a=\frac{4}{3}[/tex].
Using the [tex]\tan[/tex] difference formula, we know
[tex]\tan (a-b)=\dfrac{\tan a -\tan b}{1+\tan a\tan b}[/tex] .
Plugging in the values we know for [tex]\tan a[/tex] and [tex]\tan b[/tex], we get
[tex]\tan(a-b)=\dfrac{\frac{4}{3}-\frac{3}{4}}{1+\frac{4}{3}\cdot\frac{3}{4}}\\~~~~~~~~~~~~~~=\dfrac{\frac{7}{12}}{1+1}\\~~~~~~~~~~~~~~=\boxed{\frac{7}{24}}~.[/tex]
