Question

Assertion (A): If $x+y+z=xyz$, then $\sum \left(\frac{2x}{1-x^2}\right)=\Pi \left(\frac{2x}{1-x^2}\right)$

Reason (R): If $\tan A+\tan B+\tan C=\tan A\tan B\tan C$, then $A+B+C=n\pi ,nϵz$.

• A. A is true, R is true and R is correct explanation of A
• B. A is true, R is true and R is not correct explanation of A
• C. A is true, R is false
• D. A is false, R is true.

Answer 1

The correct option is A A is true, R is true and R is correct explanation of A

Given

$x+y+z=xyz$

Let $x=\tan A,y=\tan B,z=\tan C$

$⟹\tan A+\tan B+\tan C=\tan A\tan B\tan C$

$⟹A+B+C=n\pi$

$\sum \left(\frac{2x}{1-x^2}\right)=\Pi \left(\frac{2x}{1-x^2}\right)$

$⟹\sum \left(\frac{2\tan A}{1-{\tan }^{2}A}\right)=\Pi \left(\frac{2\tan A}{1-{\tan }^{2}A}\right)$

$⟹\sum \tan 2A=\Pi \tan 2A$

$⟹2A+2B+2C=n\pi$

$⟹A+B+C=n\pi$

Assertion and Reason are true and Reason is correct explanation.