Lazy solving!

Geometry Level 4

If none of α , β , γ , \alpha, \beta, \gamma, and δ \delta are equal to each other,then tan ⁡ ( γ − δ ) tan ⁡ ( α − β ) + tan ⁡ ( δ − β ) tan ⁡ ( α − γ ) + tan ⁡ ( β − γ ) tan ⁡ ( α − δ ) + tan ⁡ ( γ − δ ) tan ⁡ ( δ − β ) ( tan ⁡ ( β − γ ) tan ⁡ ( α − β ) tan ⁡ ( α − γ ) ( tan ⁡ ( α − δ ) = ? \dfrac{\tan(\gamma-\delta)}{\tan(\alpha-\beta)}+\dfrac{\tan(\delta-\beta)}{\tan(\alpha-\gamma)}+\dfrac{\tan(\beta-\gamma)}{\tan(\alpha-\delta)}+\dfrac{\tan(\gamma-\delta)\tan(\delta-\beta)(\tan(\beta-\gamma)}{\tan(\alpha-\beta)\tan(\alpha-\gamma)(\tan(\alpha-\delta)}\quad=\quad?

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Rohit Udaiwal by Rohit Udaiwal , 20, India

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1 solution

Akhil Bansal
Jan 7, 2016

Answer is independent of α , β , γ , δ \alpha, \beta, \gamma , \delta .
So, for quick solving, substitute α = 90 , β = 45 , γ = 45 , δ = 45 \alpha = 90 , \beta = 45 , \gamma = 45 , \delta = 45

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Did the same!

A Former Brilliant Member - 5 years, 5 months ago

Can you provide any approach to prove that the answer is independent of variables (other than bashing) ?

Venkata Karthik Bandaru - 5 years, 5 months ago

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