tan, cot in terms of p,q

Quote:

Originally Posted by daydreembelievr

Given that sin a/sin b = p and cos a/cos b = q, express tan a and cot a in terms of p and q. (a and b are acute angles). I have no idea where to start on this one. I've tried a few things but always get stuck.
Thanks much.

Here's a big start:

$\frac{\sin a}{\sin b} = p \Rightarrow \sin b = \frac{\sin a}{p}$ .... (1)

$\frac{\cos a}{\cos b} = q \Rightarrow \cos b = \frac{\cos a}{q}$ .... (2)

Square both equations and add:

$1 = \frac{\sin^2 a}{p^2} + \frac{\cos^2 a}{q^2} \Rightarrow p^2 q^2 = q^2 \sin^2 a + p^2 \cos^2 a$

$\Rightarrow \frac{p^2 q^2 }{\cos^2 a} = q^2 \tan^2 a + p^2$ .... (3)

Substitute the identity $1 + \tan^2 a = \frac{1}{\cos^2 a}$ into equation (3) and solve for $\tan^2 a$ and hence $\tan a$ .