If $\tan A=\frac{24}{7}$ and $\cos B=\frac{12}{37}$ 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{24}{7}

and

\cos B=\frac{12}{37}

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 know $\tan A = \frac{24}{7}$ , but we need to find $\tan B$ .
Find $\sin B$ : To find $\tan B$ , we need to use the Pythagorean identity $\sin^2 B + \cos^2 B = 1$ to find $\sin B$ . We know $\cos B = \frac{12}{37}$ , so we can calculate $\sin B$ . $\sin^2 B = 1 - \cos^2 B = 1 - \left(\frac{12}{37}\right)^2$ .
Calculate $\sin^2 B$ : Calculate $\sin^2 B$ . $\sin^2 B = 1 - (\frac{144}{1369}) = (\frac{1369}{1369}) - (\frac{144}{1369}) = \frac{1225}{1369}$ .
Find $\tan B$ : Find $\sin B$ by taking the square root of $\sin^2 B$ . $\newline$ Since $B$ is in Quadrant I, $\sin B$ is positive. $\newline$ $\sin B = \sqrt{\frac{1225}{1369}} = \frac{35}{37}$ .
Simplify $\tan B$ : Now we can find $\tan B$ using $\sin B$ and $\cos B$ . $\newline$ $\tan B = \frac{\sin B}{\cos B} = \frac{\frac{35}{37}}{\frac{12}{37}}$ .
Apply $\tan(A-B)$ formula: Simplify $\tan B$ . $\tan B = \left(\frac{35}{37}\right) \times \left(\frac{37}{12}\right) = \frac{35}{12}.$
Find common denominator: Now we have $\tan A$ and $\tan B$ , so we can use the $\tan(A-B)$ formula. $\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \cdot \tan B} = \frac{(\frac{24}{7}) - (\frac{35}{12})}{1 + (\frac{24}{7}) \cdot (\frac{35}{12})}$ .
Perform subtraction: Find a common denominator and subtract $\frac{24}{7}$ and $\frac{35}{12}$ . The common denominator is $7 \times 12 = 84$ . $\tan(A-B) = \frac{\left(\frac{24 \times 12}{7 \times 12} - \frac{35 \times 7}{12 \times 7}\right)}{1 + \left(\frac{24 \times 35}{7 \times 12}\right)}.$
Simplify numerator and denominator: Perform the subtraction in the numerator. $\tan(A-B) = \frac{(288 - 245) / 84}{1 + (840/84)}$ .
Divide numerator by denominator: Simplify the numerator and the term inside the parentheses in the denominator. $\tan(A-B) = \frac{43}{84} / (1 + 10)$ .
Simplify fraction: Simplify the denominator. $\tan(A-B) = \frac{43}{84} / 11$ .
Simplify fraction: Simplify the denominator. $\tan(A-B) = \frac{43}{84} / 11$ . Divide the numerator by the denominator. $\tan(A-B) = \frac{43}{84 \times 11} = \frac{43}{924}$ .
Simplify fraction: Simplify the denominator. $\tan(A-B) = \frac{43}{84} / 11$ . Divide the numerator by the denominator. $\tan(A-B) = \frac{43}{84 \times 11} = \frac{43}{924}$ . Simplify the fraction. $\tan(A-B) = \frac{43}{924}$ can be simplified by dividing both numerator and denominator by $4$ . $\tan(A-B) = \frac{43}{924} = \frac{43}{4 \times 231} = \frac{43}{924} = \frac{1}{4 \times 21} = \frac{1}{84}$ .