Question 1192632
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Find sin(2x), cos(2x), and tan(2x) from the given information.
csc(x) = 8, tan(x) < 0
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The solution by @mananth is elementary wrong.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I came to bring a correct soluition.



<pre>
You are given that csc(x) = 8, tan(x) < 0.

It means that  sin(x) = {{{1/8}}},  tan(x) < 0;  hence x is the angle in QII, second quadrant.


It implies that cos(x) = {{{-sqrt(1-sin^2(x))}}} = {{{-sqrt(1-1/64)}}} = {{{-sqrt(63/64)}}} = {{{-(3*sqrt(7))/8}}}.


Now  sin(2x) = 2*sin(x)*cos(x) = {{{2*(1/8)*((-3*sqrt(7))/8)}}} = {{{-(3*sqrt(7))/32}}};


     cos(2x) = cos^2(x) - sin^2(x) = {{{63/64}}} - {{{1/64}}} = {{{62/64}}} = {{{31/32}}}.


     tan(2x) = {{{(sin(2x))/(cos(2x))}}} = {{{-(3*sqrt(7))/31}}}.
</pre>

Solved.