TITLE: Trigonometry - How do I simplify this expression?
QUESTION [5 upvotes]: We have the expression $$ 13 \sin [ \tan ^{-1} (\dfrac{12}{5}) ] $$
Apparently the answer is 12, and I have to simplify it, and I'm assuming it means I have to show it's 12, without using a calculator.
Normally I show my own work in the questions, but in this case I have absolutely no clue how to. The only thing I know that might help is that $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$.

REPLY [4 votes]: Here is an approach. Let
$$ y=13 \sin \left( \tan ^{-1} (\dfrac{12}{5}) \right) \implies \sin \left( \tan ^{-1} (\dfrac{12}{5}) \right)=\frac{y}{13} \longrightarrow (1)$$
$$ \implies \cos \left( \tan ^{-1} (\dfrac{12}{5}) \right)=\sqrt{1-\left(\frac{y}{13}\right)^2} \longrightarrow (2). $$
Now, dividing $(1)$ by $(2)$ and simplifying gives
$$ 12^213^3-13^2y^2=0\implies y=\pm 12 \implies y=12. $$
Note: We used the identity

$$ \tan(\tan^{-1}(x))=x. $$