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If A+B+C=pi , then equality of equation (1/2)(cosA cosB + cosB cosC + cosC cosA) must hold: a) <= 3/2.   b) >=(-3)   c) <=3/8   d) >=(-1)
Asked by shanujam03 | 03 Oct, 2020, 08:59: PM Expert Answer
Given: A+B+C=pi

Consider,

(1/2)(cosA cosB + cosB cosC + cosC cosA)

= (1/4)(2cosA cosB + 2cosB cosC + 2cosC cosA)

= (1/4)[(cosA + cosB + cosC)2 - (cos2A + cos2B + cos2C)]

>= (1/4)[1 - (cos2A + cos2B + cos2C)]

As (cosA + cosB + cosC)2 >= 1

Use this inequality and get the inequality for (1/2)(cosA cosB + cosB cosC + cosC cosA)

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