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  4. If tan ((p π/4))= cot ((q π/4)) if then

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Q. If $\tan \left(\frac{p \pi}{4}\right)=\cot \left(\frac{q \pi}{4}\right)$ if then

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Solution:

$\tan \left(\frac{ p \pi}{4}\right)=\cot \left(\frac{ q \pi}{4}\right) $
$\Rightarrow \tan \left(\frac{ p \pi}{4}\right)=\tan \left(\frac{\pi}{2}-\frac{ q \pi}{4}\right) $
$\Rightarrow \frac{ p \pi}{4}= n \pi+\frac{\pi}{2}-\frac{ q \pi}{4}$
$\Rightarrow p \pi=4 n \pi+2 \pi- q \pi$
$ \Rightarrow( p + q ) \pi=2(2 n +1) \pi $
$\Rightarrow p + q =2(2 n +1)$