Statement-1: If `tanA, tanB` are the roots of the equation `x^(2)-ax-1=0,then sin^(2)(A+B)=(a^(2))/(1+a^(2))`
Statement-2: `sin^(2)(A+B)=(tan^(2)(A+B))/(1+tan^(2)(A+B))`

Options

A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement -1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

D

Explanation

We have,
`(tan^(2)theta)/(1+tan^(2)theta)=(tan^(2)theta)/(sec^(2)theta)=sin^(2)theta.`
So, statement-2 is true.
We have,
`tanA+tanB=aand tanAtanB=-1`
`thereforetan(A+B)=(tanA+tanB)/(1-tanAtanB)=a/2`
Hence, `sin^(2)(A+B)=(tan^(2)(A+B))/(1+tan^(2)(A+B))=((a^(2))/(4))/(1+(a^(2))/(4))=(a^(2))/(4+a^(2))`
So, statement-1 is not correcrt.