Question

sin[2tan(^-1)(5/12))

Answer 1

Let \(\theta=\tan^{-1}\dfrac{5}{12}\). Then use the double angle identity:
\[\sin2\theta=2\sin\theta\cos\theta=2\sin\left(\tan^{-1}\dfrac{5}{12}\right)\cos\left(\tan^{-1}\dfrac{5}{12}\right)\]
From the substitution above, you have \(\tan\theta=\dfrac{5}{12}\), which comes from the following reference triangle:

Find the missing side with the Pythagorean theorem:
\[5^2+12^2=x^2\]
Knowing the value of \(x\) will allow you to determine the cosine and since of \(\theta\), since
\[\cos\theta=\frac{12}{x}~~\text{and}~~\sin\theta=\frac{5}{x}\]
So,
\[\sin\left(2\tan^{-1}\dfrac{5}{12}\right)=2\left(\frac{5}{x}\right)\left(\frac{12}{x}\right)=\frac{120}{x^2}\]