If x - y be real- show that the equation sin 2-x2-y2 - 2 x y is possible only when x-y is not equal to zero-

Sin^2 θ=(x^2+y^2)/2xy We know, (x y)^2=x^2+y^2 2xy If (x y)^2=0⇒ x=y ⇒ x^2+y^2=2xy ⇒x^2+y^2/2xy=1 Also, x& y should not be equal to 0, otherwise it will be ...

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Standard XII

Mathematics

Properties of Argument

If x & y be r...

Question

If $x & y$ be real, show that the equation $s i n^{2} θ = (x^{2} + y^{2}) / 2 x y$ is possible only when x=y is not equal to zero.

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Solution

$s i n^{2} θ = \frac{(x^{2} + y^{2})}{2 x y}$
We know, $(x - y)^{2} = x^{2} + y^{2} - 2 x y$
If $(x - y)^{2} = 0 \Rightarrow x = y$
$\Rightarrow x^{2} + y^{2} = 2 x y$
$\Rightarrow \frac{x^{2} + y^{2}}{2 x y} = 1$
Also, $x & y$ should not be equal to 0,
otherwise it will be of the form $\frac{1}{0}$ which is not defined.

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If x&y be real, show that the equation sin2θ=(x2+y2)/2xy is possible only when x=y is not equal to zero.

sin2θ=(x2+y2)2xy We know, (x−y)2=x2+y2−2xy If (x−y)2=0⇒x=y ⇒x2+y2=2xy ⇒x2+y22xy=1 Also, x&y should not be equal to 0, otherwise it will be of the form 10 which is not defined.

If $x & y$ be real, show that the equation $s i n^{2} θ = (x^{2} + y^{2}) / 2 x y$ is possible only when x=y is not equal to zero.