If tan A = a/(a + 1) and tan B = 1/(2a + 1), Find the Value of A + B
Question:
If \[ \tan A=\frac{a}{a+1} \] and \[ \tan B=\frac{1}{2a+1} \] then the value of \[ A+B \] is
If \[ \tan A=\frac{a}{a+1} \] and \[ \tan B=\frac{1}{2a+1} \] then the value of \[ A+B \] is
Solution
We use the tangent addition formula:
\[ \tan(A+B) = \frac{\tan A+\tan B} {1-\tan A\tan B} \]
Substituting the given values,
\[ \tan(A+B) = \frac{ \frac{a}{a+1} + \frac{1}{2a+1} } { 1- \frac{a}{a+1}\cdot\frac{1}{2a+1} } \]
Taking LCM in the numerator,
\[ = \frac{ \frac{a(2a+1)+(a+1)} {(a+1)(2a+1)} } { 1- \frac{a} {(a+1)(2a+1)} } \]
\[ = \frac{ 2a^2+a+a+1 } { (a+1)(2a+1) } \div \frac{ (a+1)(2a+1)-a } { (a+1)(2a+1) } \]
\[ = \frac{ 2a^2+2a+1 } { 2a^2+2a+1 } \]
\[ =1 \]
Therefore,
\[ \tan(A+B)=1 \]
Hence,
\[ A+B=\frac{\pi}{4} \]
Final Answer
\[ \boxed{ A+B=\frac{\pi}{4} } \]
Correct Option: (d)