If = a a+1 and B=1 2a+1 , then the value of A + B is: - Vidyadip

MCQ

If $\tan\text{A}=\frac{\text{a}}{\text{a}+1}$ and $\text{B}=\frac{1}{2\text{a}+1},$ then the value of $A + B$ is:

A
$0$
B
$\frac{\pi}{2}$
C
$\frac\pi3$
✓
$\frac\pi4$

✓

Answer

Correct option: D.

$\frac\pi4$

$\tan(\text{A+B})=\frac{\tan\text{A}+\tan\text{B}}{1-\tan\text{A}\tan\text{B}}$
$=\frac{\frac{\text{a}}{\text{a}+1}+\frac{1}{2\text{a}+1}}{1-\frac{\text{a}}{(\text{a}+1)(2\text{a}+1)}}$
$=\frac{2\text{a}^2+\text{a}+\text{a}+1}{2\text{a}^2+3\text{a}+1-\text{a}}$
$=\frac{2\text{a}^2+2\text{a}+1}{2\text{a}^2+2\text{a}+1}$
$=1$
$\therefore \text{ A+B}=\tan^{-1}(1)=\frac\pi4.$

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