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Question

$text If end text sin text end text x cos space y equals 1 fourth$ and $3 tan space x equals 4 tan space y$ then $sin space left parenthesis x minus y right parenthesis equals$

$\frac{1}{16}$
$\frac{7}{16}$
$\frac{3}{4}$
$\frac{3}{16}$

hint Hint:

First we will convert the expression $3 tan space x equals 4 tan space y$ in terms of $sin open parentheses theta close parentheses$ and $cos open parentheses theta close parentheses$ . Then we will use the formula of $sin open parentheses x minus y close parentheses equals sin open parentheses x close parentheses cos open parentheses y close parentheses minus sin open parentheses y close parentheses cos open parentheses x close parentheses$ to find the value of $sin open parentheses x minus y close parentheses$ .

The correct answer is: $1 over 16$

In this question we are given expression $sin open parentheses x close parentheses cos open parentheses y close parentheses equals 1 fourth$ and $3 tan space x equals 4 tan space y$ and we have to find the value of $sin space left parenthesis x minus y right parenthesis$ .
Step1: Rearranging the term $3 tan space x equals 4 tan space y$
We know that $tan open parentheses theta close parentheses equals fraction numerator sin open parentheses theta close parentheses over denominator cos open parentheses theta close parentheses end fraction$ . So, we can rewrite the above expression as
$fraction numerator 3 sin open parentheses x close parentheses over denominator cos open parentheses x close parentheses end fraction equals fraction numerator 4 sin open parentheses y close parentheses over denominator cos open parentheses y close parentheses end fraction$
Step2: Cross multiplying the terms.
$3 sin open parentheses x close parentheses cos open parentheses y close parentheses equals 4 sin open parentheses y close parentheses cos open parentheses x close parentheses$ and we are already given the value of $sin open parentheses x close parentheses cos open parentheses y close parentheses$ as $1 fourth$
=> $sin open parentheses y close parentheses cos open parentheses x close parentheses equals 3 over 16$
Step3: Using the formula of $sin open parentheses x minus y close parentheses equals sin open parentheses x close parentheses cos open parentheses y close parentheses minus sin open parentheses y close parentheses cos open parentheses x close parentheses$
By putting $sin open parentheses y close parentheses cos open parentheses x close parentheses equals 3 over 16$ and $sin open parentheses x close parentheses cos open parentheses y close parentheses equals 1 fourth$ in the above expression we get
$sin open parentheses x minus y close parentheses equals 1 fourth minus 3 over 16$
=> $sin open parentheses x minus y close parentheses equals fraction numerator 4 minus 3 over denominator 16 end fraction$
=> $sin open parentheses x minus y close parentheses equals 1 over 16$

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