Question

If A + B + C = π(A, B, C > 0) and the ∠C is obtuse, then ______.

Options

• tan A tan B > 1

• tan A tan B < 1

• tan A tan B = 1

• None of these

Answer 1

If A + B + C = π(A, B, C > 0) and the ∠C is obtuse, then tan A tan B < 1.

Explanation:

A + B + C = π

A + B = π – C

`\implies` tan(A + B) = tan(π – C)

`\implies (tan A + tan B)/(1 - tan A tan C)` = tan(π – C)

`\implies (tan A + tan B)/(1 - tan A tan C)` = – tan C

Now, C is an obtuse angle, hence

`\implies` tan C < 0

`\implies` – tan C > 0

`\implies (tan A + tan B)/(1 - tan A tan B) > 0`

`\implies` 1 – tan A tan B > 0  ...[∵ A, B are acute angles; ∴ tan A > 0, tan B > 0]

`\implies` tan A tan B < 1