Funny thing. Multiplying both the sides by 0? Alright this maybe really funny but I want to know why is this wrong. We often come across identities which we prove by multiplying both the sides of the identity by a certain entity but why don't we multiply it by $0$. That way every identity will be proved in one single line. That is so stupid. I mean, by that way we may also say that $1=2=3$. I know it is wrong. But why? I mean if we can multiply both the sides by $2$ then why not by $0$.  For example, consider the following trigonometric identity :

Prove the identity : $\sin^2 \theta = \tan^2 \theta \cos ^2 \theta$

Usual way

To Prove : $\sin^2 \theta = \tan^2 \theta \cos ^2 \theta $

$\displaystyle \implies {\sin^2 \theta \over \cos^2 \theta } = {\tan^2 \theta \cos ^2 \theta \over \cos^2 \theta}$ (multiplying both the sides by $\displaystyle 1 \over \cos^2 \theta$)

$\implies \tan ^2 \theta = \tan^2\theta$

$\implies LHS=RHS$

$\therefore proved$

Funny way

To Prove : $\sin^2 \theta = \tan^2 \theta \cos ^2 \theta $

$\displaystyle \implies {\sin^2 \times 0} = {\tan^2 \theta \cos ^2 \theta \times 0}$ (multiplying both the sides by $0$)

$\implies 0 = 0$

$\therefore proved$

Please explain why is this wrong.
$a = b$ implies $ac = bc$, but $ac = bc$ doesn't imply $a = b$. (Not immediately. Read below.)

The way you usually get $a = b$ from $ac=bc$ is by multiplying both sides with $1/c$, which is only available when $c \ne 0$.
Zero differs from other numbers and mathematical entities which represent some sort of value because it is the absence of value, not a value itself. Thus, it would be incorrect to say $a \cdot 0 = b \cdot 0$ means that $a = b$. Similarly, in the case of infinity: let's say that $a$ does equal $b$. However, $a \cdot \infty \neq b \cdot \infty$ because infinity does not have a definite or specific value. Rather, it is a concept of something without limit.