Question

Disprove the given non-identity by giving a counter-example,
Square root of sin^2 x-9 = sin x-3

Answer 1

\[\sqrt{\sin^2x-9}=\sin x-3~~?\]

Answer 2

Yes that's the equation

Answer 3

Consider the fact that the right side, under the square root, can be factored:
\[\sqrt{(\sin x+3)(\sin x-3)}\]
Is this equal to \(\sin x-3\)?

Answer 4

*left side, not right

Answer 5

i think you meant
sqrt[ sin^2(x-3) ]

Answer 6

consider also that the left hand side is an imaginary number, since \(\sin^2(x)-9\leq -8\)

Answer 7

Notice:
\[\large\begin{align*}\sqrt{(\sin x+3)(\sin x-3)}&\overbrace{=}^{?}\sin x-3\\
\sqrt{(\sin x+3)(\sin x-3)}&\overbrace{=}^{?}\sqrt{(\sin x-3)(\sin x-3)}\\
\sqrt{(\sin x+3)}\sqrt{(\sin x-3)}&\overbrace{=}^{?}\sqrt{(\sin x-3)}\sqrt{(\sin x-3)}\\
\sqrt{\sin x+3}&\overbrace{=}^{?}\sqrt{\sin x-3}\\
3&~\not=-3\end{align*}\]
@satellite73 is quite right about what s/he said, and this is exactly why. No such number \(x\) will satisfy the equation.

Answer 8

Ok thank you so much very helpful

Answer 9

yw