MHB Solve a trigonometric equation

[anemone]

Let $y$ be in radians and $0<y<\dfrac{\pi}{4}$.

Solve for $y $ if $\tan 4y=\dfrac{\cos y-\sin y}{\cos y +\sin y}$.

 

[Saitama]

Let $y$ be in radians and $0<y<\dfrac{\pi}{4}$.

Solve for $y $ if $\tan 4y=\dfrac{\cos y-\sin y}{\cos y +\sin y}$.

Spoiler

We can rewrite the RHS as:
$$\tan 4y=\frac{1-\tan y}{1+\tan y}=\tan\left(\frac{\pi}{4}-y\right)$$
$$\Rightarrow 4y=n\pi+\frac{\pi}{4}-y$$
Only n=0 gives a solution in the specified range, hence
$$y=\frac{\pi}{20}$$

 

[I like Serena]

$$\frac{1-\tan y}{1+\tan y}=\tan\left(\frac{\pi}{4}-y\right)$$

How did you get that?
It's not something you had to learn by heart did you?

 

[Saitama]

It's not something you had to learn by heart did you?

Nope. :D

I used the following formula:
$$\tan(a-b)=\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}$$
with $a=\pi/4$ and $b=y$. :)