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The value of tan 1° tan 2° tan 3° tan 89° is equal to
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The correct answer is:
$1$
\begin{align*}
&\text{Given} \\
&f(x)=\begin{vmatrix} 1 & x & x & x+1 \\ 2x & x(x-1) & (x+1)x \\ 3x(x-1) & x(x-1)(x-2) & (x + 1) \cdot x(x - 1) \end{vmatrix} \\ \\
&\text{On applying } C_{3} \to C_{3} - C_{2} \text{ we get } \\ \\
&f(x)=\begin{vmatrix} 1 & x & 1 \\ 2x & x(x-1) & 2x \\ 3x(x-1) & x(x-1)(x-2) & 3x(x - 1) \end{vmatrix}=0. \\
&f(x) = 0 \Rightarrow f(100) = 0
\end{align*}
&\text{Given} \\
&f(x)=\begin{vmatrix} 1 & x & x & x+1 \\ 2x & x(x-1) & (x+1)x \\ 3x(x-1) & x(x-1)(x-2) & (x + 1) \cdot x(x - 1) \end{vmatrix} \\ \\
&\text{On applying } C_{3} \to C_{3} - C_{2} \text{ we get } \\ \\
&f(x)=\begin{vmatrix} 1 & x & 1 \\ 2x & x(x-1) & 2x \\ 3x(x-1) & x(x-1)(x-2) & 3x(x - 1) \end{vmatrix}=0. \\
&f(x) = 0 \Rightarrow f(100) = 0
\end{align*}
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