What is the value of $\tan(A)/\tan(B)$?

Question

here is the the question :

http://upimage.us/server/php/files/mathquestionsnew222.png

http://upimage.us/server/php/files/mathquestionsnew222.png

I tried to come up with the idea of this question but all of my attempts were failed :(

How is this question solved??

Answer 1

$$\dfrac{\tan(A)}{\tan(B)} = \dfrac{\sin(A)}{\cos(A)}\cdot \dfrac{\cos(B)}{\sin(B)} = \dfrac{\cos^2(B)}{\sin^2(B)} = \cot^2(B) = \csc^2(B) - 1$$

Answer 2

Note that $tan \:\theta$ is defined as $\dfrac{sin\:\theta}{cos\:\theta}$

$\therefore \dfrac{tan\:(A)}{tan\:(B)} = \dfrac{\dfrac{sin(A)}{cos(A)}}{\dfrac{sin(B)}{cos(B)}} = \dfrac{\cos^2(B)}{\sin^2(B)} = \cot^2(B) = \csc^2(B) - 1$