question_answer

In a triangle \[ABC\], right angled at C, the value of \[\tan A+\tan B\] is [Pb. CET 1990; Karnataka CET 1999; MP PET 2001]

A) \[a+b\]

B) \[\frac{{{a}^{2}}}{bc}\]

C) \[\frac{{{b}^{2}}}{ac}\]

D) \[\frac{{{c}^{2}}}{ab}\]

Correct Answer: D

Solution :

Given, A right-angled triangle ABC with right angled at C. Let a, b and c be the lengths of sides BC, CA and AB respectively. We know from the Pythagoras theorem that \[{{c}^{2}}={{a}^{2}}+{{b}^{2}}\]and\[\tan A=\frac{a}{b}.\] Similarly, \[\tan B=\frac{b}{a}.\]  Therefore, \[\tan A+\tan B=\frac{a}{b}+\frac{b}{a}=\frac{{{a}^{2}}+{{b}^{2}}}{ab}=\frac{{{c}^{2}}}{ab}.\]