Question
For positive acute angles $A$ and $B$, it is known that $\tan A=\frac{9}{40}$ and $\cos B=\frac{28}{53}$. Find the value of $\cos (A-B)$ in simplest form.
Answer
Final Answer: The value of \(\cos (A-B)\) in simplest form is \(\boxed{0.7017947537965946}\).
Step 1 ：We are given that \(\tan A = \frac{9}{40}\) and \(\cos B = \frac{28}{53}\).
Step 2 ：We know that \(\cos^2 A = 1 - \sin^2 A\) and \(\sin^2 B = 1 - \cos^2 B\).
Step 3 ：Since \(\tan A = \frac{\sin A}{\cos A}\), we can find \(\sin A\) and \(\cos A\) using the Pythagorean identity. Similarly, we can find \(\sin B\) using the Pythagorean identity.
Step 4 ：Using the given values, we find that \(\sin A = 0.21951219512195125\), \(\cos A = 0.975609756097561\), and \(\sin B = 0.8490566037735848\).
Step 5 ：We can substitute these values into the formula for \(\cos (A-B)\) to find the answer.
Step 6 ：\(\cos (A-B) = \cos A \cos B + \sin A \sin B = 0.7017947537965946\).
Step 7 ：Final Answer: The value of \(\cos (A-B)\) in simplest form is \(\boxed{0.7017947537965946}\).
