If $\tan A=\frac{12}{35}$ and $\cos B=\frac{15}{17}$ and angles A and B are in Quadrant I, find the value of $\tan (A-B)$ . $\newline$ Answer:

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Q. If

\tan A=\frac{12}{35}

and

\cos B=\frac{15}{17}

and angles A and B are in Quadrant I, find the value of

\tan (A-B)

\newline

Answer:

Use $\tan(A-B)$ Formula: Use the formula for $\tan(A-B)$ , which is $\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \cdot \tan B}$ . We are given $\tan A = \frac{12}{35}$ . We need to find $\tan B$ using the given $\cos B = \frac{15}{17}$ .
Find $\tan B$ : Since $\cos B = \frac{15}{17}$ , we can find $\sin B$ using the Pythagorean identity $\sin^2 B + \cos^2 B = 1$ . $\newline$ $\sin^2 B = 1 - \cos^2 B$ $\newline$ $\sin^2 B = 1 - \left(\frac{15}{17}\right)^2$ $\newline$ $\sin^2 B = 1 - \frac{225}{289}$ $\newline$ $\sin^2 B = \frac{289 - 225}{289}$ $\newline$ $\sin^2 B = \frac{64}{289}$ $\newline$ $\sin B = \sqrt{\frac{64}{289}}$ $\newline$ $\cos B = \frac{15}{17}$ $0$ , since $\cos B = \frac{15}{17}$ $1$ is in Quadrant I, $\sin B$ is positive.
Calculate $\tan B$ : Now we can find $\tan B$ using the ratio $\tan B = \frac{\sin B}{\cos B}$ . $\tan B = \frac{8}{17} / \frac{15}{17}$ $\tan B = \frac{8}{17} \cdot \frac{17}{15}$ $\tan B = \frac{8}{15}$
Substitute tan Values: Substitute the values of $\tan A$ and $\tan B$ into the $\tan(A-B)$ formula. $\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \cdot \tan B}$ $\tan(A-B) = \frac{(\frac{12}{35}) - (\frac{8}{15})}{1 + (\frac{12}{35}) \cdot (\frac{8}{15})}$
Simplify Numerator/Denominator: Simplify the numerator and the denominator separately. $\newline$ Numerator: $(\frac{12}{35}) - (\frac{8}{15}) = (\frac{12\times3 - 8\times7}{35\times3})$ $\newline$ Numerator: $(\frac{36 - 56}{105}) = \frac{-20}{105} = \frac{-4}{21}$ $\newline$ Denominator: $1 + (\frac{12}{35}) \times (\frac{8}{15}) = 1 + (\frac{96}{(35\times15)})$ $\newline$ Denominator: $1 + (\frac{96}{525}) = (\frac{525 + 96}{525}) = \frac{621}{525} = \frac{207}{175}$
Divide to Find $\tan(A-B)$ : Now divide the numerator by the denominator to find $\tan(A-B)$ . $\newline$ $\tan(A-B) = \frac{-4}{21} / \frac{207}{175}$ $\newline$ $\tan(A-B) = \frac{-4}{21} \cdot \frac{175}{207}$
Final Simplification: Simplify the expression by canceling out common factors. $\newline$ $\tan(A-B) = (-4 \times 175) / (21 \times 207)$ $\newline$ $\tan(A-B) = (-700) / (4347)$ $\newline$ $\tan(A-B) = -100 / 621$