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If $|\begin{matrix} A & D & 1 \\ B & E & 1 \\ C & F & 1 \end{matrix}| = |\begin{matrix} π & \frac{π}{2} & 3 \\ B - C & E - F & 0 \\ C & F & 1 \end{matrix}|$ then
Statement 1: $\tan A + \tan B + \tan C = \tan A . \tan B . \tan C$
Statement 2: $\cot D + \cot E + \cot F = \cot D . \cot E . \cot F$ Then

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a

only one is true

b

only 2 is true

c

both 1 and 2 are true

d

None of them is true

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detailed solution

Correct option is C

If A+B+C=π⇒tanA+tanB+tanC=tanAtanB·tanC If D+E+F=π2 then cotD+cotE+cotF=cotD·cotE·cotFNow, AD1BE1CF1 apply R1→R1+R2+R3, R2→R2-R3 = A+B+CD+E+F3B-CE-F0CF1 = ππ23B-CE-F0CF1 given ⇒A+B+C=π and D+E+F=π2 ∴Statement (1) and statement (2) both are true.

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