find the exact value of $\cos (\alpha+\beta)$ if $\sin \alpha$ is $\frac{-4}{5}$ and $\sin \beta=\frac{-5}{13}$ with $\alpha$ in quadrant IV and $\beta$ in quadrant III

Step 1 ：We are given that \(\sin \alpha = -\frac{4}{5}\) and \(\sin \beta = -\frac{5}{13}\), with \(\alpha\) in quadrant IV and \(\beta\) in quadrant III.

Step 2 ：We need to find the exact value of \(\cos (\alpha+\beta)\), which is given by the formula \(\cos \alpha \cos \beta - \sin \alpha \sin \beta\).

Step 3 ：To find \(\cos \alpha\) and \(\cos \beta\), we can use the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\) and \(\sin^2 \beta + \cos^2 \beta = 1\).

Step 4 ：Since \(\alpha\) is in quadrant IV, \(\cos \alpha\) is positive. Solving the Pythagorean identity for \(\alpha\), we find that \(\cos \alpha = 0.6\).

Step 5 ：Similarly, since \(\beta\) is in quadrant III, \(\cos \beta\) is negative. Solving the Pythagorean identity for \(\beta\), we find that \(\cos \beta = -0.9230769230769231\).

Step 6 ：Substituting these values into the formula for \(\cos (\alpha+\beta)\), we find that \(\cos (\alpha+\beta) = -0.8615384615384615\).

Step 7 ：Final Answer: The exact value of \(\cos (\alpha+\beta)\) is \(\boxed{-0.8615384615384615}\).