give the exact value of the expression without using a calculator. sin(2 tan^(-1)(-3/4)) sin(2…

Explanation:

Step1: Let $\theta=\tan^{-1}(-\frac{3}{4})$

Then $\tan\theta = -\frac{3}{4}$, and we consider a right - triangle where the opposite side $y=- 3$ and the adjacent side $x = 4$ (since $\tan\theta=\frac{y}{x}$). By the Pythagorean theorem, the hypotenuse $r=\sqrt{x^{2}+y^{2}}=\sqrt{4^{2}+(-3)^{2}} = 5$. So, $\sin\theta=-\frac{3}{5}$ and $\cos\theta=\frac{4}{5}$.

Step2: Use the double - angle formula for sine

The double - angle formula for sine is $\sin(2\alpha)=2\sin\alpha\cos\alpha$. Here, $\alpha = \tan^{-1}(-\frac{3}{4})$, so $\sin(2\tan^{-1}(-\frac{3}{4}))=2\sin(\tan^{-1}(-\frac{3}{4}))\cos(\tan^{-1}(-\frac{3}{4}))$. Substituting $\sin\theta = -\frac{3}{5}$ and $\cos\theta=\frac{4}{5}$ into the double - angle formula, we get $2\times(-\frac{3}{5})\times\frac{4}{5}$.

Step3: Calculate the result

$2\times(-\frac{3}{5})\times\frac{4}{5}=-\frac{24}{25}$.

Answer:

$-\frac{24}{25}$