Question

Prove that $x$ & $y$ are not odd multiple of $\frac{\pi }{2}$ then $\tan x=\tan y\Rightarrow x=n\pi +y$, where $n\in z$

Answer 1

If $x$ and $y$ are not odd multiple at $\frac{\pi }{2}$ then,

$\tan x-\tan y\Rightarrow x=n\pi +yn\in z$

Proof:-

Given that

$\tan x=\tan y$

$\begin{array}{c}\frac{\sin x}{\cos x}=\frac{siny}{cosy}\\ \Rightarrow \sin x\cos y=\cos x\sin y\\ \Rightarrow \sin x\cos y-\cos x\sin y=0\\ \Rightarrow \sin \left(x-y\right)=0\left[\begin{array}{c}\because \sin \theta =0\\ \Rightarrow \theta =n\pi \end{array}\right]\\ \Rightarrow x-y=n\pi \\ \Rightarrow x=n\pi +y\end{array}$

Where,

$n\pm 1,\pm 2$.