Question

I don't need you to work it out, just theorem(s) i need to reach the answer :)

Answer 1

Not sure why such an old question is showing up on my feed...

Anyway, let $x=\tan^{-1}\dfrac43$ and $y=\sin^{-1}\dfrac35$. Then we want to find the exact value of $\cos(x-y)$.

Use the angle difference identity:

$\cos(x-y)=\cos x\cos y+\sin x\sin y$

and right away we find $\sin y=\dfrac35$. By the Pythagorean theorem, we also find $\cos y=\dfrac45$. (Actually, this could potentially be negative, but let's assume all angles are in the first quadrant for convenience.)

Meanwhile, if $\tan x=\dfrac43$, then (by Pythagorean theorem) $\sec x=\dfrac53$, so $\cos x=\dfrac35$. And from this, $\sin x=\dfrac45$.

So,

$\cos\left(\tan^{-1}\dfrac43-\sin^{-1}\dfrac35\right)=\dfrac35\cdot\dfrac45+\dfrac45\cdot\dfrac35=\dfrac{24}{25}$