Question

Find the value of $\tan 1\tan 2....\tan 89$

• A. $-1$
• B. $1$
• C. $\sqrt{2}$
• D. $2$

Answer 1

The correct option is B $1$

$\tan 1\cdot \tan \mathrm{2.....}\tan 89$

but $\tan 89$ can be written as $\cot 1\left[\tan \right(90-1)=\cot 1]$ by $\tan (90-\theta )=\cot \left(\theta \right)$

likewise...,

$\tan 88=\cot 2$

$\tan 87=\cot 3$

....... up to $\tan 46=\cot 44$

then middle one is $\tan 45=1$

so it is

$\tan 1\cdot \tan \mathrm{2....}\tan 44\cdot \tan 45\cdot \cot \mathrm{44.....}\cot 1$

$\tan$ and $\cot$ cancel out by$\left[\tan \right(\theta {)}^{\ast }\cot \left(\theta \right)=1]$

so, remaining is $1$

answer$=1$.