tan(pi-x) Formula | Simplify tan(x-π)

The tan(π-x) formula is given by tan(π-x)= -tanx. The formula of tan(pi-θ) is equal to tan(π-θ)= -tanθ. In this post, we will learn how to compute tan(pi-x) and tan(θ-pi).

Note that

• tan(π-x)= -tanx or, tan(π-θ)= -tanθ
• tan(x-π)= tanx or, tan(θ-π)= tan θ.

Proof of tan(pi-x) Formula

Let us use the formula:

tan(a-b) = $\dfrac{\tan a -\tan b}{1+\tan a \tan b}$ …(∗)

Put a=π, b=x. Thus, we get that

tan(π-x) = $\dfrac{\tan \pi -\tan x}{1+\tan \pi \tan x}$

= $\dfrac{0 -\tan x}{1+0 \cdot \tan x}$ as the value of tanπ = 0.

= $\dfrac{-\tan x}{1}$

= -tanx

So the formula of tan(π-x) is equal to -tanx, that is,

tan(π-x) = -tanx

tan(π-θ) Formula: In the above formula, if we replace x by θ, we will get the formula of tan(π-θ) which is as follows:

tan(π-θ) = -tanθ

Proof of tan(x-pi) Formula

In the above formula (∗) of tan(a-b), let us put

a=x, b=π.

So, tan(x-π)

= $\dfrac{\tan x- \tan \pi}{1+\tan x \tan \pi}$

= $\dfrac{\tan x- 0}{1+\tan x \cdot 0}$ as tan(π) =0.

= tanx

So the formula of tan(x-π) is equal to tan(x-π)= tanx.

tan(θ-π) Formula: In the above formula, replacing x by θ, we get the following: the formula of tan(theta-π) is given by tan(θ-π)= tanθ.

More Formulas:

Formula of sin(π-x)

Value of sin(5π/2)

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

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