The number of points inside (or) on the circle x2 + y2 = 4 satisfying tan4⁡x+cot4⁡x+1=3sin2⁡y is _____

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The number of points inside (or) on the circle x² + y² = 4 satisfying $\tan^{4} x + \cot^{4} x + 1 = 3 \sin^{2} y is ______$

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answer is 4.

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Detailed Solution

$\begin{array}{l} \frac{\tan^{4} x + \frac{1}{\tan^{4} x}}{2} \geq \sqrt{\tan^{4} x \cdot \frac{1}{\tan^{4} x}} \Rightarrow \tan^{4} x + \cot^{4} x + 1 \geq 3 \\ \Rightarrow 3 \sin^{2} y \leq 3 \Rightarrow \sin^{2} y = 1 a n d \tan^{4} x = 1 \\ \Rightarrow y = \pm \frac{π}{2} \Rightarrow x = \pm \frac{π}{4}, \pm \frac{3 π}{4} \\ ∴ P o i n t s = (\frac{π}{4}, \frac{π}{2}), (\frac{π}{4}, \frac{- π}{2}), (\frac{- π}{4}, \frac{π}{2}), (- \frac{π}{4}, - \frac{π}{2}), (\frac{3 π}{4}, \frac{π}{2}), (- \frac{3 π}{4}, \frac{π}{2}), (\frac{3 π}{4}, - \frac{π}{2}), (- \frac{3 π}{4}, - \frac{π}{2}) \\ b u t (\frac{3 π}{4}, \frac{π}{2}), (- \frac{3 π}{4}, \frac{π}{2}), (\frac{3 π}{4}, - \frac{π}{2}), (- \frac{3 π}{4}, - \frac{π}{2}) d o e s n o t l i e i n s i d e t h e c i r c l e x^{2} + y^{2} = 4 \end{array}$ Required number of points =4