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3.Trigonometrical Ratios, Functions and Identities
hard
જો સમીકરણ ${\sin ^2}\theta = \frac{{{x^2} + {y^2}}}{{2xy}},x,y, \ne 0$ શકય હોય તો
A
$x = y$
B
$x = \, -y$
C
$2x = y$
D
એકપણ નહીં
Solution
Now, $\sin ^{2} \theta=\frac{x^{2}+y^{2}}{2 x y}$
$\therefore \mathrm{x},$ $y$ have same sign
$\frac{x^{2}+y^{2}}{2 x y}=\frac{1}{2}\left[(\sqrt{\frac{x}{y}}-\sqrt{\frac{y}{x}})^{2}+2\right] \geq 1$
Now, $\sin ^{2} \theta \leq 1 .$
Therefore, $\frac{x^{2}+y^{2}}{2 x y}=1$
$ \Rightarrow x=y$
Standard 11
Mathematics
