Solve equation for general solution $2 (sin x)^{2} + (sin 2 x)^{2} = 2$ .

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Solution

$2 {sin}^{2} x + (sin 2 x)^{2} = 2$

$2 {sin}^{2} x + (2 sin x cos x)^{2} = 2$

$2 {sin}^{2} x + 4 {sin}^{2} x (1 - {sin}^{2} x) = 2$

$2 {sin}^{2} x + 4 {sin}^{2} x - 4 {sin}^{4} x = 2$

$4 {sin}^{4} x - 6 {sin}^{2} x + 2 = 0$

Solving the above quadratic equation, we get,

${sin}^{2} x = \frac{1}{2}$ and ${sin}^{2} x = 1$

$sin x = \pm \frac{1}{2}$ and $sin x = \pm 1$

So, $x = (2 n + 1) \frac{π}{2}$ and $x = (2 n + 1) \frac{π}{2}$

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c

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