Solve: $4 sin x + cos x = 4.$

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Solution

$cos x = 4 (1 - sin x)$
${cos}^{2} \frac{x}{2} - {sin}^{2} \frac{x}{2} = 4 {(cos \frac{x}{2} - sin \frac{x}{2})}^{2}$
$\frac{1 + tan \frac{x}{2}}{1 - tan \frac{x}{2}} = 4$
So, $t a n (\frac{π}{2} + \frac{x}{2}) = 4, x = 2 n π + {2 tan}^{- 1} (4) - \frac{π}{2}$ and $t a n \frac{x}{2} = 1, x = 2 n π + \frac{π}{2} .$

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