Question

The value of $\tan 1\tan 2\tan 3\tan 4...\tan 89$

• A. 1
• B. -1
• C. 0
• D. 2

Answer 1

The correct option is A 1

$\tan 1\tan 2\tan 3\tan 4...\tan 89$

$=\tan (90-89)\tan (90-88)\tan (90-87)\tan (90-86)...\tan 45.....\tan 86\tan 87\tan 88\tan 89$

There are $\frac{89}{2}=44$ terms $+1$ term

That $1$ term is the middle term$=\tan 45=1$

and other terms will form the reciprocals

$=\cot 89\cot 88\cot 87\cot 86...\tan 45.....\tan 86\tan 87\tan 88\tan 89$

$=\left(\tan 89\cot 89\right)\times \left(\tan 88\cot 88\right)\times \left(\tan 87\cot 87\right)\times \left(\tan 86\cot 86\right)\times ...\tan 45$ by re-arranging

$=1\times 1\times 1\times \mathrm{1...1}=1$