Question

\[ \tan1^\circ \tan2^\circ \tan3^\circ \cdots \tan89^\circ \]

\[ \text{is equal to} \]

Solution

Using identity

\[ \tan\theta \tan(90^\circ-\theta)=1 \]

\[ \tan1^\circ \tan89^\circ=1 \]

\[ \tan2^\circ \tan88^\circ=1 \]

\[ \tan3^\circ \tan87^\circ=1 \]

Similarly all pairs are equal to \(1\).

Also,

\[ \tan45^\circ=1 \]

Therefore,

\[ \tan1^\circ \tan2^\circ \tan3^\circ \cdots \tan89^\circ=1 \]

Answer

\[ \boxed{1} \]