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Question

if $tan space left parenthesis A minus B right parenthesis equals 7 divided by 24 comma tan space A equals 4 divided by 3$ where A, B acute, then A+B=

$\frac{π}{5}$
$\frac{π}{4}$
$\frac{π}{3}$
$\frac{π}{2}$

hint Hint:

We will use the formula of $tan open parentheses A minus B close parentheses equals fraction numerator tan open parentheses A close parentheses minus tan open parentheses B close parentheses over denominator 1 plus tan open parentheses A close parentheses tan open parentheses B close parentheses end fraction$ to find $tan open parentheses B close parentheses$ and then we will use the formula of $tan open parentheses A plus B close parentheses equals fraction numerator tan open parentheses A close parentheses plus tan open parentheses B close parentheses over denominator 1 minus tan open parentheses A close parentheses tan open parentheses B close parentheses end fraction$ to find the value of A+B

The correct answer is: $pi over 2$

In this question we are given expression $tan space left parenthesis A minus B right parenthesis equals 7 divided by 24$ and $tan open parentheses A close parentheses equals 4 divided by 3$ and it is also given that A and B are acute angle and we have to find the value of A+B.
Step1: Using the formula of $tan open parentheses A minus B close parentheses$
We know that $tan open parentheses A minus B close parentheses equals fraction numerator tan open parentheses A close parentheses minus tan open parentheses B close parentheses over denominator 1 plus tan open parentheses A close parentheses tan open parentheses B close parentheses end fraction$ . By putting the values given to us in the expression of $tan open parentheses A minus B close parentheses$ .
$7 over 24 equals fraction numerator begin display style 4 over 3 end style minus tan open parentheses B close parentheses over denominator 1 plus 4 over 3 tan open parentheses B close parentheses end fraction$
Step2: Rearranging the equation we get.
$7 over 24 plus 7 over 24 cross times 4 over 3 tan open parentheses B close parentheses equals 4 over 3 minus tan open parentheses B close parentheses$
=> $7 over 24 minus 4 over 3 equals negative 7 over 18 tan open parentheses B close parentheses minus tan open parentheses B close parentheses$
=> $fraction numerator 21 minus 96 over denominator 72 end fraction equals fraction numerator negative 25 tan open parentheses B close parentheses over denominator 18 end fraction$
=> $fraction numerator negative 75 over denominator 72 end fraction equals fraction numerator negative 25 tan open parentheses B close parentheses over denominator 18 end fraction$
$tan open parentheses B close parentheses equals 3 over 4$
Step3: Finding the values of A+B
In order to find the value of A+B we will use the formula of $tan open parentheses A plus B close parentheses$
We know that $tan open parentheses A plus B close parentheses equals fraction numerator tan open parentheses A close parentheses plus tan open parentheses B close parentheses over denominator 1 minus tan open parentheses A close parentheses tan open parentheses B close parentheses end fraction$
Since, $1 minus tan open parentheses A close parentheses tan open parentheses B close parentheses space equals space 1 minus 4 over 3 cross times 3 over 4$ is $0$ .
The value of $tan open parentheses A plus B close parentheses equals 1 over 0$ which means
A+B = $pi over 2$ .

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