Use the Double Argument Property color(red)(tan 2theta=(2tan theta)/(1-tan^2 theta)
Since color(red)(cos theta is positive it means that color(red)(theta is either in quadrant I or IV. I will find both and you can pick the answer according to the right quadrant that is given in the problem.
Here it is,
color(purple)(Quadrant I.
If color(red)(cos theta = 5/13 then color(red)(adjacent = 5 and hypote n use = 13 therefore by using pythagorean property we have color(red)(opposite = 12. Therefore, color(red)(tan theta=(opposite)/(adjacent)=12/5
Now put it in to the formula for color(red)(tan 2 theta. That is,
color(red)((2tan theta)/(1-tan^2 theta)=color(blue)((2(12/5))/(1-(12/5)^2
=color(blue)((24/5)/(1-144/25)
=color(blue)((24/5)/((25-144)/25)
=color(blue)((24/5)/(-119/25
=color(blue)(24/5 * -25/119
=color(blue)(-120/119
color( Purple)(Quadrant IV
If color(red)(cos theta = 5/13 then color(red)(adjacent = 5 and hypote n use = 13 therefore the color(red)(opposite = -12. Thus, color(red)(tan theta=-12/5
color(red)((2tan theta)/(1-tan^2 theta)=color(blue)((2(-12/5))/(1-(12/5)^2
=color(blue)((-24/5)/(1-144/25)
=color(blue)((-24/5)/((25-144)/25)
=color(blue)((-24/5)/(-119/25
=color(blue)(24/5 * 25/119
=color(blue)(120/119
