# Is there any intuitive way to simplify $\frac {\sin x} {\cos 3x} +\frac{\sin 3x}{\cos 9x} +\frac{\sin 9x}{\cos 27 x}$?

$$\frac {\sin x} {\cos 3x}+ \frac{\sin 3x}{\cos 9x} +\frac{\sin 9x}{\cos 27 x}$$

This question does have answers here Show that $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$

And also here and here : https://math.stackexchange.com/a/1299968

Concept of Trigonometric identities

However, all of the answers in these questions are the same method; the method that I have given below.

I was working with this expression for a long time, and finally, I looked it up online.

The only way I saw online to simplify this is by:- $$\frac {\sin x} {\cos 3x}=\frac {\sin x} {\cos 3x} * \frac {\cos x} {\cos x}$$ $$= \frac {\sin x \cos x} {\cos 3x\cos x}$$ $$= \frac{\sin 2x}{2*\cos 3x \cos x}$$ $$=\frac 1 2 *\frac {\sin 2x} {\cos 3x \cos x}$$ $$=\frac 1 2 *\frac{\sin (3x-x)}{\cos 3x \cos x}$$ $$=\frac 1 2 * \frac {\sin 3x\cos x -\cos 3x\sin x} {\cos 3x\cos x}$$ $$= \frac 1 2 * (\tan 3x - \tan x)$$

So you first have to know to multiply this expression with $$\frac {\cos x} {\cos x}$$, then you have to realise that $$\sin x\cos x = \frac {\sin 2x} 2$$. and then you have to realise that $$\sin 2x= \sin (3x-x)$$.

This is very unintuitive. The first thing anyone not familiar with this would try to do is to convert $$\cos 3x$$ in terms of $$\cos x$$, and then work from there.

So this is my question. Is there any intuitive way to work out this expression?