Question

The value of $(\tan 1°\tan 2°\tan 3°\cdots \cdots \cdots \cdots \cdots \cdots \tan 89°)$ is

• A. $0$
• B. $1$
• C. $2$
• D. $\frac{1}{2}$

Answer 1

The correct option is B

$1$

Use the trigonometric identity $\tan (90-\theta )=cot\theta$ and $\tan \theta =\frac{1}{cot\theta }$

Here we have ,

$$\begin{gathered} \tan 1°\tan 2°\tan 3°\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \tan 89° \\ =\tan 1°\tan 2°\tan 3°\cdots \cdots \cdots \cdots \cdots \tan 45°\tan 46°\tan 47°\cdots \cdots \tan 88°\tan 89° \end{gathered}$$

We know that $\text{tan45°=1}$

$$\begin{gathered} \tan 1°\tan 2°\tan 3°\cdots \cdots \cdots \tan 44°.1.\tan 46°\tan 47°\cdots \cdots \tan 88°\tan 89° \\ =\tan 1°\tan 2°\tan 3°\cdots \cdots \tan 44°.1.\tan (90-44)°\tan (90-43)°\cdots \cdots \tan (90-2)°\tan (90-1)° \end{gathered}$$

We also know that $\tan (90-\theta )=cot\theta$ and $\tan \theta =\frac{1}{cot\theta }$

$$\begin{gathered} \tan 1°\tan 2°\tan 3°\cdots \cdots \cdots \tan 44°.1.cot44°cot43°cot42°\cdots cot3°cot2°cot1° \\ =\tan 1°\tan 2°\tan 3°\cdots \cdots \tan 44°.1.\frac{1}{\tan 44°}.\frac{1}{\tan 43°}.\frac{1}{\tan 42°}\cdots \frac{1}{\tan 3°}.\frac{1}{\tan 2°}.\frac{1}{\tan 1°} \\ =1 \end{gathered}$$

Hence option(2) is the correct answer and all other options are incorrect answers.