If x+y+z=π, then prove tanx + tany + tanz = tanx tany tanz

In this post, we will prove tanx + tany + tanz = tanx tany tanz when x+y+z is equal to π. Here we will use the following formula:

tan(x+y) = $\dfrac{\tan x+\tan y}{1-\tan x \tan y}$.

Question: If x+y+z=π, then prove that tanx + tany + tanz = tanx tany tanz.

Solution:

Given that x+y+z=π.

⇒ x+y = π-z

So tan(x+y) = tan(π-z)

⇒ $\dfrac{\tan x+\tan y}{1-\tan x \tan y}$ = – tan z

⇒ tanx + tany = – tanz (1- tanx tany)

⇒ tanx + tany = – tanz + tanx tany tanz

⇒ tanx + tany + tanz = tanx tany tanz.

Hence, we have shown that if x+y+z=π, then tanx + tany + tanz = tanx tany tanz.

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