Evaluate : $cos 5 x + cos 3 x = 0$

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Solution

$c o s 5 x + c o s 3 x = 0$
$\Rightarrow 2 c o s [\frac{5 x + 3 x}{2}] c o s [\frac{5 x - 3 x}{2}] = 0$
$\Rightarrow 2 c o s 4 x c o s x = 0$
$\Rightarrow c o s x = 0$ or $c o s 4 x = 0$
$\Rightarrow x \in \frac{n π}{2}$ or $4 x \in \frac{n π}{2} n \in N$
$\Rightarrow x \in \frac{n π}{8} n \in N$
Solution set of cos $5 x + c o s 3 x = 0$ is
$x = \frac{n π}{8}$

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\int \frac{{cos}^{3} x + {cos}^{5} x}{{sin}^{2} x + {sin}^{4} x} d x

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\int (\frac{{cos}^{3} x + {cos}^{5} x}{{sin}^{2} x + {sin}^{4} x}) d x =

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