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Equations involving sinx + cosx and sinx.cosx

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Question

Find the general solution:
$s e c^{2} 2 x = 1 - t a n 2 x$

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Solution

$s e c^{2} 2 x = 1 - t a n 2 x$
$\Rightarrow 1 + {tan}^{2} 2 x = 1 - t a n 2 x$
$\Rightarrow tan 2 x (tan 2 x + 1) = 0$
$\Rightarrow tan 2 x = 0, - 1$
$\Rightarrow 2 x = n π, n π - \frac{π}{4}$
$\Rightarrow x = \frac{n π}{2}, \frac{n π}{2} - \frac{π}{8}$

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