If sin 2 = 4 x y ( x + y ) 2 is true, if and only if

Question

If $\sin^{2} θ = \frac{4 x y}{{(x + y)}^{2}}$ is true, if and only if

Smart Achievers · Accepted Answer

(3)

$∵$ $\sin^{2} θ \leq 1$

$∴$ $\frac{4 x y}{{(x + y)}^{2}} \leq 1$

$\Rightarrow$ $x^{2} + y^{2} + 2 x y - 4 x y \geq 0$

$\Rightarrow$ ${(x - y)}^{2} \geq 0$

which is true for all real values of $x$ and $y$ , provided $x + y \neq 0$ , otherwise $\frac{4 x y}{{(x + y)}^{2}}$ will be meaningless.

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