Solve the following equations.
$c o s 3 x t a n 5 x = s i n 7 x .$

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Solution

$cos 3 x \times tan 5 x = sin 7 x => cos 3 x \times \frac{sin 5 x}{cos 5 x} = sin 7 x => cos 3 x \times sin 5 x = sin 7 x \times cos 5 x => sin 5 x \times cos 3 x = sin 7 x \times cos 5 x$
Using $sin A + sin B = 2 sin (\frac{A + B}{2}) \times cos (\frac{A - B}{2})$
We can write
$sin 8 x + sin 2 x = sin 12 x + sin 2 x => sin 12 x - sin 8 x = 0 => 2 sin 2 x cos 10 x = 0$
$sin 2 x = 0 cos 10 x = 0 => 2 x = n π => 10 x = \frac{(2 m + 1) π}{2} => x = \frac{n π}{2} => x = \frac{(2 m + 1) π}{20}$
$m, n$ belongs to integers

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