2015/035) If $5\csc(x) - 2\cot(x)= 5$ then find the value of $5\cot(x) - 2\csc(x)$.

Let $5 \csc\, x - 2 \cot\, x = 5 \cdots (1)$

Then, squaring both sides

$25 \csc^2 x + 4 \cot^2 x - 20 \csc\, x \cot\, x = 25$

or $25 ( \csc^2x - 1 ) + 4 ( \csc^2 x - 1 ) - 20 \csc\, x \cot\, x = 0$

or $25 \cot^2 x + 4 \csc^2 x - 20 \csc\, x \cot\, x = 4$

or $(5\cot\, x - 2 \csc\, x )^2 = 4$

or $5\cot\, x - 2 \csc\, x = \pm 2$