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.
Best Answer
$a = b$ 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 $c \ne 0$.