Let $\alpha \in (0,\frac{\pi }{2})$, be constant.If the integral $\int \frac{\mathrm{t}\mathrm{a}\mathrm{n}x+\mathrm{t}\mathrm{a}\mathrm{n}\alpha }{\mathrm{t}\mathrm{a}\mathrm{n}x-\tan \alpha }dx=A\left(x\right)\mathrm{c}\mathrm{o}\mathrm{s}2\alpha +B\left(x\right)\mathrm{s}\mathrm{i}\mathrm{n}2\alpha +C$, where C is a constant of integration, then the functions $A\left(x\right)$ and $B\left(x\right)$ are respectively
 

• Correct answer
  A

  $x-\alpha$ and $\mathrm{l}\mathrm{o}{\mathrm{g}}_e\left|\mathrm{s}\mathrm{i}\mathrm{n}\left(x-\alpha \right)\right|$
• B

  $x+\alpha$ and ${\mathrm{l}\mathrm{o}\mathrm{g}}_e\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(x-\alpha \right)\right|$
• C

  $x+\alpha$ and ${\mathrm{l}\mathrm{o}\mathrm{g}}_e\left|\mathrm{s}\mathrm{i}\mathrm{n}\left(x+\alpha \right)\right|$
• D

  $x-\alpha$ and ${\mathrm{l}\mathrm{o}\mathrm{g}}_e\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(x-\alpha \right)\right|$

Hence, the correct answer is option: 'A'.