Trigonometry Question

Question

Show that if α, β and γ are the angles of a triangle, then we would have but
tanα+tanβ+tanγ=tanαtanβtanγ

I need help with this. I've been trying but getting nowhere. Thanks.

Answer 1

This is called the triple tangent identity. In a triangle, α+β+γ=π

So γ=π-α-β

tan(γ) = tan(π-α-β)

=-tan(α+β)

=-tan(α)-tan(β)

  1-tan(α)tan(β)

=tan(α)+tan(β)

  tan(α)tan(β)-1

(tan(α)tan(β)-1)tan(γ) = tan(α)+tan(β)

tan(α)tan(β)tan(γ)-tan(γ) = tan(α)+tan(β)

tan(α)tan(β)tan(γ) = tan(α)+tan(β)+tan(γ)