# Solve a trigonometric equation

#### anemone

##### MHB POTW Director
Staff member
Let $y$ be in radians and $0<y<\dfrac{\pi}{4}$.

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

#### Pranav

##### Well-known member
Let $y$ be in radians and $0<y<\dfrac{\pi}{4}$.

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

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
$$y=\frac{\pi}{20}$$

#### Klaas van Aarsen

##### MHB Seeker
Staff member
$$\frac{1-\tan y}{1+\tan y}=\tan\left(\frac{\pi}{4}-y\right)$$
How did you get that?
It's not something you had to learn by heart did you?

#### Pranav

##### Well-known member
It's not something you had to learn by heart did you?
Nope.

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