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Question

$text If end text 0 less than alpha comma beta less than pi over 4 comma cos space left parenthesis alpha plus beta right parenthesis equals 4 over 5 comma sin space left parenthesis alpha minus beta right parenthesis equals 5 over 13 text then end text tan text end text 2 alpha equals$

$\frac{33}{56}$
$\frac{56}{33}$
$\frac{16}{63}$
$\frac{14}{63}$

hint Hint:

In this question first we will find the values of $sin open parentheses alpha plus beta close parentheses$ and $cos open parentheses alpha minus beta close parentheses$ and then use the formula of $sin open parentheses A plus B close parentheses equals sin open parentheses A close parentheses cos open parentheses B close parentheses plus cos open parentheses A close parentheses sin open parentheses B close parentheses$ to find the value of $sin open parentheses 2 alpha close parentheses$ by putting A = $alpha plus beta$ and B = $alpha minus beta$ . Then at last we will find the value of $tan open parentheses 2 alpha close parentheses$ from $sin open parentheses 2 alpha close parentheses$ .

The correct answer is: $56 over 33$

In this question we are given expression $cos space left parenthesis alpha plus beta right parenthesis equals 4 over 5$ and $sin space left parenthesis alpha minus beta right parenthesis equals 5 over 13$ and we have to find the value of $tan open parentheses 2 alpha close parentheses$
Step1: First we will find the value of $sin open parentheses 2 alpha close parentheses$ by using the formula of $sin open parentheses A plus B close parentheses$ .
We know that $sin open parentheses A plus B close parentheses equals sin open parentheses A close parentheses cos open parentheses B close parentheses plus cos open parentheses A close parentheses sin open parentheses B close parentheses$ ,
Let's assume A = $alpha plus beta$ and B = $alpha minus beta$ then we get A+B = $2 alpha$ .
$sin open parentheses left parenthesis alpha plus beta right parenthesis plus left parenthesis alpha minus beta right parenthesis close parentheses equals sin open parentheses alpha plus beta close parentheses cos open parentheses alpha minus beta close parentheses plus cos open parentheses alpha plus beta close parentheses sin open parentheses alpha minus beta close parentheses$
Step2: Finding the value of $sin open parentheses alpha plus beta close parentheses$ and $cos open parentheses alpha minus beta close parentheses$
We also know that
$sin open parentheses alpha plus beta close parentheses equals square root of 1 minus cos open parentheses alpha plus beta close parentheses squared end root$
=> $sin open parentheses alpha plus beta close parentheses equals square root of 1 minus left parenthesis 4 over 5 right parenthesis squared end root$
=> $sin open parentheses alpha plus beta close parentheses equals 3 over 5$
Similarly $cos open parentheses alpha minus beta close parentheses equals 12 over 13$
Now we will put all the values in the formula $sin open parentheses left parenthesis alpha plus beta right parenthesis plus left parenthesis alpha minus beta right parenthesis close parentheses equals sin open parentheses alpha plus beta close parentheses cos open parentheses alpha minus beta close parentheses plus cos open parentheses alpha plus beta close parentheses sin open parentheses alpha minus beta close parentheses$ .
$sin open parentheses 2 alpha close parentheses equals 3 over 5 cross times 12 over 13 plus 4 over 5 cross times 5 over 13$
=> $sin open parentheses 2 alpha close parentheses space equals space 56 over 65$
Step3: Finding $tan open parentheses 2 alpha close parentheses$ from $sin open parentheses 2 alpha close parentheses$ .
We know that $tan open parentheses 2 alpha close parentheses equals fraction numerator sin open parentheses 2 alpha close parentheses over denominator square root of 1 minus sin open parentheses 2 alpha close parentheses squared end root end fraction$
From above equation we get $tan open parentheses 2 alpha close parentheses equals 56 over 33$

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