Tangent Differences

Geometry Level 3

tan ⁡ 7 5 ∘ − tan ⁡ 3 0 ∘ − tan ⁡ 3 0 ∘ tan ⁡ 7 5 ∘ = ? \tan75^\circ - \tan30^\circ - \tan30^\circ \tan75^\circ = \, ?

2 tan ⁡ 7 5 ∘ \tan 75^\circ 1 tan ⁡ 1 5 ∘ \tan 15 ^ \circ

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Shovoñ Shovon by Shovoñ shovon , 24, Bangladesh

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2 solutions

Chew-Seong Cheong
Oct 18, 2016

@Shovoñ shovon 's solution in LaTex:

tan ⁡ 7 5 ∘ − tan ⁡ 3 0 ∘ 1 + tan ⁡ 7 5 ∘ tan ⁡ 3 0 ∘ = tan ⁡ ( 7 5 ∘ − 3 0 ∘ ) = tan ⁡ 4 5 ∘ = 1 ⟹ tan ⁡ 7 5 ∘ − tan ⁡ 3 0 ∘ = 1 + tan ⁡ 7 5 ∘ tan ⁡ 3 0 ∘ tan ⁡ 7 5 ∘ − tan ⁡ 3 0 ∘ − tan ⁡ 3 0 ∘ tan ⁡ 7 5 ∘ = 1 \begin{aligned} \frac {\tan 75^\circ - \tan 30^\circ}{1+ \tan 75^\circ \tan 30^\circ} & = \tan (75^\circ - 30^\circ) \\ & = \tan 45^\circ \\ & = 1 \\ \implies \tan 75^\circ - \tan 30^\circ & = 1+ \tan 75^\circ \tan 30^\circ \\ \tan 75^\circ - \tan 30^\circ - \tan 30^\circ \tan 75^\circ & = \boxed{1} \end{aligned}

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Shovoñ Shovon
Oct 18, 2016

tan(75°-30°)=(tan75°-tan30°)/(1+tan75°.tan30°) or,1+tan30°.tan75°=tan75°-tan30° or,tan75°-tan30°-tan30°.tan75°=1

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