If tan A = 5/6 and tan B = 1/11, Prove that A + B = π/4

Question

If \[ \tan A=\frac{5}{6} \] and \[ \tan B=\frac{1}{11} \] prove that:

\[ A+B=\frac{\pi}{4} \]

Proof

Using the identity:

\[ \tan(A+B) = \frac{\tan A+\tan B} {1-\tan A\tan B} \]

Substituting the given values:

\[ \tan(A+B) = \frac{\frac{5}{6}+\frac{1}{11}} {1-\left(\frac{5}{6}\times\frac{1}{11}\right)} \]

Taking LCM in the numerator:

\[ = \frac{\frac{55+6}{66}} {1-\frac{5}{66}} \]

\[ = \frac{\frac{61}{66}} {\frac{61}{66}} \]

\[ =1 \]

Therefore,

\[ \tan(A+B)=1 \]

We know that:

\[ \tan\frac{\pi}{4}=1 \]

Hence,

\[ A+B=\frac{\pi}{4} \]

Hence proved.