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The sides a, b, c (in that order) of triangle ABC form an arithmetic progression. Three angles alpha, beta, gamma are defined by cos(alpha) = a/(b+c), cos(beta) = b/(c+a), cos(gamma) = c/(a+b). Find the value of tan²(alpha/2) + tan²(gamma/2). (All symbols carry their usual meaning in triangle ABC.)

1. 1
2. 1/2
3. 1/3
4. 2/3

Correct answer: 2/3

Solution

Since a, b, c are in AP, a+c = 2b, so a+b+c = 3b. Applying the half-angle identity to each cosine and adding gives (b+c-a + a+b-c)/(a+b+c) = 2b/(3b) = 2/3.

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