Question

$\mathrm{If}\tan \left(\frac{p\pi }{4}\right)=cot\left(\frac{\mathrm{q\pi }}{4}\right),\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}(p+q)\mathrm{is}$
.

• A. $2(2n+1),n\in Z.$
• B. $(2n-1),n\in Z.$
• C. $2n+\frac{1}{2},n\in Z.$
• D. $n+\frac{1}{2},n\in Z.$

Answer 1

The correct option is A $2(2n+1),n\in Z.$

Here, the given equation can be written as:
$\tan \left(\frac{p\pi }{4}\right)=\cot \left(\frac{\mathrm{q\pi }}{4}\right)\Rightarrow \tan \left(\frac{p\pi }{4}\right)=\tan \left(\frac{\pi }{2}-\frac{\mathrm{q\pi }}{4}\right)\mathrm{Now},\mathrm{the}\mathrm{general}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{above}\mathrm{equation}\mathrm{can}\mathrm{be}\mathrm{given}\mathrm{as}:\frac{p\pi }{4}=n\pi +\frac{\pi }{2}-\frac{\mathrm{q\pi }}{4},n\in Z.\Rightarrow \frac{\mathrm{p\pi }}{4}+\frac{\mathrm{q\pi }}{4}=\left(n+\frac{1}{2}\right)\pi ,n\in Z.\Rightarrow \left(p+q\right)\pi =\frac{4(2n+1)}{2}\pi ,n\in Z.\Rightarrow \left(p+q\right)=2(2n+1),n\in Z.$