give the exact value of the expression without using a calculator. cos(2 arctan(-3/4)) cos(2 arctan(-3/4)) =…

Explanation:

Step1: Let $\theta = \arctan(-\frac{3}{4})$

Then $\tan\theta=-\frac{3}{4}$, and we know that $\tan\theta=\frac{y}{x}$, so we can consider a right - triangle in the coordinate plane where $y = - 3$ and $x = 4$. By the Pythagorean theorem, $r=\sqrt{x^{2}+y^{2}}=\sqrt{4^{2}+(-3)^{2}} = 5$. So, $\sin\theta=-\frac{3}{5}$ and $\cos\theta=\frac{4}{5}$.

Step2: Use the double - angle formula for cosine

The double - angle formula for cosine is $\cos(2\theta)=\cos^{2}\theta-\sin^{2}\theta$. Substitute $\sin\theta = -\frac{3}{5}$ and $\cos\theta=\frac{4}{5}$ into the formula: $\cos(2\theta)=(\frac{4}{5})^{2}-(-\frac{3}{5})^{2}$.

Step3: Calculate the value

$(\frac{4}{5})^{2}-(-\frac{3}{5})^{2}=\frac{16}{25}-\frac{9}{25}=\frac{16 - 9}{25}=\frac{7}{25}$.

Answer:

$\frac{7}{25}$