Question

$\text{If tan A and tan B are the roots of}ab{x}^2-c^{2}x+ab=0\text{where a, b, c are the sides of the triangle ABC, then the value of}si{n}^{2}A+si{n}^{2}B+si{n}^{2}Cis$

• A. 1
• B. 3
• C. 4
• D. 2

Answer 1

The correct option is D

2

$\text{Given tan A and tan B are the roots of the equation}ab{x}^2-c^{2}x+ab=0\therefore tanA+tanB=\frac{c^2}{ab},tanAtanB=1\Rightarrow A+B=\frac{\pi }{2}\Rightarrow C=\frac{\pi }{2}\therefore si{n}^{2}A+si{n}^{2}B+si{n}^{2}C=si{n}^2(\frac{\pi }{2}-B)+si{n}^{2}B+si{n}^2\frac{\pi }{2}co{s}^{2}B+si{n}^{2}B+1=2$