The value of tan 55°cot 35°+ cot 1° cot 2° cot 3° …. cot 90°, is
Question:

The value of $\frac{\tan 55^{\circ}}{\cot 35^{\circ}}+\cot 1^{\circ} \cot 2^{\circ} \cot 3^{\circ}$ $\cot 90^{\circ}$, is

(a) −2
(b) 2
(c) 1
(d) 0

Solution:

We have to find the value of the following expression

$\frac{\tan 55^{\circ}}{\cot 35^{\circ}}+\cot 1^{\circ} \cot 2^{\circ} \cot 3^{\circ} \ldots \cot 90^{\circ}$

$=\frac{\tan 55^{\circ}}{\cot 35^{\circ}}+\cot 1^{\circ} \cot 2^{\circ} \cot 3 \ldots \cot 90^{\circ}$

$=\frac{\tan \left(90^{\circ}-35^{\circ}\right)}{\cot 35^{\circ}}+\cot \left(90^{\circ}-89^{\circ}\right) \cot \left(90^{\circ}-88^{\circ}\right) \cot \left(90^{\circ}-87^{\circ}\right) \ldots \cot 87 \cot 88^{\circ} \cot 89^{\circ} \ldots \cot 90^{\circ}$

$=\frac{\cot 35^{\circ}}{\cot 35^{\circ}}+\tan 89^{\circ} \tan 88^{\circ} \tan 87^{\circ} \ldots \cot 87 \cot 88^{\circ} \cot 89^{\circ} \ldots \cot 90^{\circ}$

$=1+1 \times 1 \times 1 \ldots \times 0$

$=1$

As $\cot 90^{\circ}=0$

Hence the correct option is $(c)$

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