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Geometry Level 3

$\cot { 32^\circ } \tan { 86^\circ } \tan { 23^\circ } \tan { 16^\circ } \tan { 67^\circ } \tan { 4^\circ } \tan { 74^\circ } \tan { 32^\circ }$ Find the value of the trigonometric expression above.

\sqrt { 3 }

1

Undefined

\frac { 1 }{ \sqrt { 3 } }

1 solution

Chew-Seong Cheong
Jan 17, 2017

$\begin{aligned} P & = \cot 32^\circ {\color{#3D99F6} \tan 86^\circ \tan 23^\circ \tan 16^\circ} \tan 67^\circ \tan 4^\circ \tan 74^\circ \tan 32^\circ & \small \color{#3D99F6} \text{As } \tan \theta = \cot (90^\circ - \theta) \\ & = \cot 32^\circ {\color{#3D99F6} \cot 4^\circ \cot 67^\circ \cot 74^\circ} \tan 67^\circ \tan 4^\circ \tan 74^\circ \tan 32^\circ & \small \color{#3D99F6} \text{Note that } \cot \theta = \frac 1{\tan \theta} \\ & = \frac {\tan 67^\circ \tan 4^\circ \tan 74^\circ \tan 32^\circ}{\color{#3D99F6} \tan 32^\circ \tan 4^\circ \tan 67^\circ \tan 74^\circ} \\ & = \boxed{1} \end{aligned}$

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