Find out mathematical flaw...

Que: Prove that sin^4 A - sin^2 A cos^2 A +cos^4 A + 3 sin^2 A cos^2 A - (sin^2 A + cos^2 A) + sin^3 A cosec^2 A = sin A. Student: Multiply R.H.S. & L.H.S. by 0 i.e. 0=0. Teacher gave marks: 10/10 * 0 = 0. For you: Can you find out the mathematical flaw in the answer given by the student and the marks given by the teacher which have given us misleading results? NOTE: We often multiply both sides by a rational no. in such questions but if we do that with 0 it becomes misleading. What's the reason?

#Geometry

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

I think the reason why multiplying by $0$ is misleading is that you cannot go back, i.e. dividing by $0$ is not allowed. That follows the same concept when squaring a number. For example, given a negative number, say $-a$ for some positive real number $a^2$ . If you square it, it would become $a$ but you cannot say that getting the square root of $a^2$ will lead you to $a=-a$ . I will post later for the proof and for those who cannot understand, it says: $\sin^4 A - \sin^2 A \cos^2 A +\cos^4 A + 3 \sin^2 A \cos^2 A - (\sin^2 A + \cos^2 A) + \sin^3 A \csc^2 A = \sin A$ Hint: Sum of Two Squares and Pythagorean Theorem P.S. I thought that student was smart but then I realized that the teacher was smarter. :P

Marc Vince Casimiro - 6 years, 6 months ago

$\sin^4 A - \sin^2 A \cos^2 A +\cos^4 A + 3 \sin^2 A \cos^2 A - (\sin^2 A + \cos^2 A) + \sin^3 A \csc^2 A = \sin A$ $(\sin^2A)^2+(\cos^2A)^2+ 2 \sin^2 A \cos^2 A - 1 + \sin^3 A \csc^2 A=\sin A$ $(\sin^2A+cos^2A)^2-1+\sin^3 A \csc^2 A=\sin A$ $\sin^3A\csc^2A=\sin A$ $\sin^3A*\frac{1}{\sin^2A}=\sin A$ $\sin A=\sin A$

Marc Vince Casimiro - 6 years, 6 months ago

But 0 is not -ve. Then how can you apply the concept of squaring to this problem.

Sudo Jarvis - 6 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$