Not defined?

Geometry Level 2

tan ( 8 5 ) tan ( 17 5 ) tan ( 26 5 ) tan ( 35 5 ) = ? \large \tan (85^\circ) \tan (175^\circ) \tan (265^\circ) \tan (355^\circ) = ?

\infty Undefined 1 0

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1 solution

Chew-Seong Cheong
Oct 22, 2017

X = tan 8 5 tan 17 5 tan 26 5 tan 35 5 = tan 8 5 tan 17 5 tan ( 18 0 + 8 5 ) tan ( 18 0 + 17 5 ) Note that tan ( 18 0 + θ ) = tan θ = tan 2 8 5 tan 2 17 5 = tan 2 8 5 tan 2 ( 18 0 5 ) = tan 2 8 5 tan 2 ( 5 ) = tan 2 8 5 ( tan 5 ) 2 = tan 2 8 5 tan 2 5 = tan 2 ( 9 0 5 ) tan 2 5 Note that tan ( 9 0 θ ) = cot θ = cot 2 5 tan 2 5 Note that cot θ = 1 tan θ = tan 2 5 tan 2 5 = 1 \begin{aligned} X & = \tan 85^\circ \tan 175^\circ \tan 265^\circ \tan 355^\circ \\ & = \tan 85^\circ \tan 175^\circ \color{#3D99F6} \tan (180^\circ + 85^\circ) \tan (180^\circ + 175^\circ) & \small \color{#3D99F6} \text{Note that }\tan (180^\circ+\theta) = \tan \theta \\ & = \tan^2 85^\circ \tan^2 175^\circ \\ & = \tan^2 85^\circ \tan^2 (180^\circ - 5^\circ) \\ & = \tan^2 85^\circ \tan^2 (- 5^\circ) \\ & = \tan^2 85^\circ (-\tan 5^\circ)^2 \\ & = \tan^2 85^\circ \tan^2 5^\circ \\ & = \tan^2 (90^\circ - 5^\circ)\tan^2 5^\circ & \small \color{#3D99F6} \text{Note that }\tan (90^\circ-\theta) = \cot \theta \\ & = \cot^2 5^\circ \tan^2 5^\circ & \small \color{#3D99F6} \text{Note that }\cot \theta = \frac 1{\tan \theta} \\ & = \frac {\tan^2 5^\circ} {\tan^2 5^\circ} \\ & = \boxed{1} \end{aligned}

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