Question

Observe the following statements

$\left(i\right)In\Delta ABC,bco{s}^2\frac{C}{2}+cco{s}^2\frac{B}{2}=s$

$\left(ii\right)In\Delta ABC,cot\frac{A}{2}=\frac{b+c}{a}\Rightarrow B={90}^0$

Which of the following is correct?

• A. Both I and II are true
• B. I is true, II is false
• C. I is false, II is true
• D. Both I and II are false

Answer 1

The correct option is A

Both I and II are true

$\left(i\right)Now,bco{s}^2\frac{C}{2}+cco{s}^2\frac{B}{2}=b.\frac{s(s-c)}{ab}+c.\frac{s(s-b)}{ac}=\frac{s}{a}(s-c+s-b)=\frac{s}{a}(2s-(b+c\left)\right)=\frac{s}{a}.a=s\therefore StatementIistrue.$

$\left(II\right)Letcot\frac{A}{2}=\frac{b+c}{a}\frac{cos\frac{A}{2}}{sin\frac{A}{2}}=\frac{sinB+sinC}{sinA}\Rightarrow \frac{cos\frac{A}{2}}{sin\frac{A}{2}}=\frac{2sin\left(\frac{B+C}{2}\right)cos\left(\frac{B-C}{2}\right)}{2sin\frac{A}{2}cos\frac{A}{2}}\Rightarrow \frac{A}{2}=\frac{B-C}{2}\Rightarrow A+C=BButA+B+C=\pi ,\Rightarrow B=\frac{\pi }{2}\therefore StatementIIistrue.\text{Hence,option (a) is the correct answer.}$