Question

Statement $1:$ If $xy+yz+zx=1$ where $x,y,z\in R^{+}$, then
$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}=\frac{2}{\sqrt{(1+x^2)(1+y^2)(1+z^2)}}$

Statement $2:$ In a triangle $ABC$, $\sin 2A+\sin 2B+\sin 2C=4\sin A\sin B\sin C$

• A. Both the statements are TRUE and Statement $2$ is the correct explanation of Statement $1$
• B. Both the statements are TRUE but Statement $2$ is NOT the correct explanation of Statement $1$
• C. Statement $1$ is TRUE and Statement $2$ is FALSE
• D. Statement $1$ is FALSE and Statement $2$ is TRUE

Answer 1

The correct option is A Both the statements are TRUE and Statement $2$ is the correct explanation of Statement $1$

Let $x=\cot A,y=\cot B,z=\cot C$
$\Rightarrow \sum \cot A\cot B=1$
$\Rightarrow A+B+C=n\pi$

Statement $1:$
LHS $=\frac{1}{2}[\sin 2A+\sin 2B+\sin 2C]$
$=2\sin A\sin B\sin C$

RHS $=\frac{2}{|\text{cosec}A\text{cosec}B\text{cosec}C|}$
$=\pm 2\sin A\sin B\sin C$

Statement $2$ is true as it is a conditional identity.