\(A\), \(B\) and \(C\) are the three angles of a triangle. What is the value of \(\tan\left(\displaystyle\frac{A+B}{2}\right)\cdot\tan\left(\displaystyle\frac{C}{2}\right)\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 1
The angles of a triangle add up to \(180^\circ\). Halving this, \(\frac{A+B}{2}\) and \(\frac{C}{2}\) add up to \(90^\circ\), so they are complementary, and \(\tan(90^\circ-x)=\cot x\).
\(A+B=180^\circ-C\), so \(\displaystyle\frac{A+B}{2}=90^\circ-\frac{C}{2}\)
\(\tan\left(\displaystyle\frac{A+B}{2}\right)=\tan\left(90^\circ-\displaystyle\frac{C}{2}\right)=\cot\displaystyle\frac{C}{2}\)
Product \(=\cot\displaystyle\frac{C}{2}\cdot\tan\frac{C}{2}=1\)