$tan \,(5\pi \,cos\, \theta ) = cot\, (5 \pi\, sin\, \theta )$

$tan \,(5\pi \,cos\, \theta ) = tan\, \left( {\frac{\pi }{2}\, – \,5\pi \sin \theta } \right)$

$5\pi \,cos\theta = n\pi + \pi /2 – 5\pi \,sin\theta$

$(cos\theta + sin\theta ) = \left( {\frac{{2n + 1}}{{10}}} \right)$

$\Rightarrow -1 < \frac{{2n + 1}}{{10\sqrt 2 }} < 1$ 

$\Rightarrow$ $ – \,\frac{{10\sqrt 2 \, – 1}}{2}$ $< n <$ $\frac{{10\sqrt 2 \, – 1}}{2}$

$\therefore$ $n= 14$ for each $‘n’$ there are two values of $\theta$

no. of solutions $= 28$