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Question
The value of `tan A/2 tan B/2 + tan B/2 tan C/2 + tan C/2 tan A/2` is ______.
Options
3
1
5
4
MCQ
Fill in the Blanks
Solution
The value of `tan A/2 tan B/2 + tan B/2 tan C/2 + tan C/2 tan A/2` is 1.
Explanation:
In ΔABC, A + B + C = π
∴ A + B = π – C
∴ `(A + B)/2 = (π - C)/2 = π/2 - C/2`
`tan(A/2 + B/2) = tan(π/2 - C/2)`
∴ `(tan A/2 + tan B/2)/(1 - tan A/2 tan B/2) = cot C/2`
∴ `(tan A/2 + tan B/2)/(1 - tan A/2 tan B/2) = 1/(tan C/2)`
∴ `[tan A/2 + tan B/2] tan C/2 = 1 - tan A/2 tan B/2`
∴ `tan A/2 tan C/2 + tan B/2 tan C/2 = 1 - tan A/2 tan B/2`
∴ `tan A/2 tan C/2 + tan B/2 tan C/2 + tan A/2 tan B/2` = 1
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Factorization Formulae - Trigonometric Functions of Angles of a Triangle
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