Question

cos(A+B+C)+cos (-A+B+C)+cos (A-B+C) + cos (A+B-C) sin (A+B+C)+sin (- A+B+C) + sin (A − B+C) -sin (A+B-C) = cot C​

Answer 1

Answer:

SOLUTION

A + B + C =π

                A + B = π - C

                tan (A + B = tan (π - C)

tanA+tanB1−tanA.tanB = -tanC

tan A + tan B + tan C = tan A tan B tan C                 - - - - - - (1)

For three values tan A, tan B & tan C

AM ≥ GM

tanA+tanB+tanC3 ≥ (tanA.tanB.tanC)13

[tanA + tanB + tanC = tanA  tanB  tanC]

 (tanA.tanB.tanC)23≥3

(1cotA.cotB.cotC)23≥3

                                             {cotA.cotB.cotC = k}

(1k)23≥3

1k≥(3)32

k ≤1(3)32

k ≤13.312

k ≤13√3

Answer 2

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Refer to the attachment mate