If $x=\cot6^\circ\cot42^\circ$ and $y=\tan66^\circ\tan78^\circ$, then determine the ratio of $x$ and $y$

Question

If $x=\cot6^\circ\cot42^\circ$ and $y=\tan66^\circ\tan78^\circ$, then

A) $2x=y$

B) $x=2y$

C) $x=y$

D) $2x=3y$

I seriously don’t know where to start. I don’t need the complete answer, but a starting statement which would give me the direction to solve it would be helpful.

Thanks a lot.

Answer 1

as @Ross said $x = \cot 6 \tan 48 , y = \cot 24 \cot 12$ then we look at

$$\frac{x}{y} = \dfrac{\cot 6 \tan 48 }{\cot 24 \cot 12} = \frac{\cot 6 \tan 24}{\cot 48 \cot 12}$$

$$\cot 6 \tan 24 = \frac{\cos 6 \sin 24}{\sin 6 \cos 24}= \frac{\sin 30 + \sin 18}{\sin 30 - \sin 18} = \frac{1 + 2\sin 18 }{1 - 2\sin 18} $$

I used $$\sin a \cos b = 0.5 ( \sin (a+b) + \sin (a-b)) \\ \cos a \sin b = 0.5 ( \sin (a+b) - \sin (a-b))$$

You need also

$$\cos a \cos b = 0.5 ( \cos (a+b) + \cos (a-b) ) \\ \sin a \sin b = 0.5 ( \cos (a-b) - \cos (a+b))$$ to simplify the Denominator can you proceed ?