tan(x-y) Formula, Proof | tan(x-y) Identity

The tan(x-y) is the tangent of the difference of the angles x and y. Tan(x-y) formula/identity is given as follows:

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

In this post, we will prove the formula of tan x-y.

Proof of tan(x-y) Formula

tan(x-y) = ${\displaystyle \frac{\tan x-\tan y}{1+\tan x\tan y}}$

Proof:

We need the below two formulas:

sin(x-y) = sinx cosy – cosx siny

cos(x-y) = sinx siny + cosx cosy

To prove the tan(x-y) formula, we will proceed as follows. Note that $\tan \theta =\frac{\sin \theta }{\cos \theta }$. Thus, we have

tan(x-y) = ${\displaystyle \frac{\sin (x-y)}{\cos (x-y)}}$

⇒ tan(x-y) = ${\displaystyle \frac{\sin x\cos y–\cos x\sin y}{\sin x\sin y+\cos x\cos y}}$

Dividing both the numerator and the denominator by cosx cosy, we obtain that

tan(x-y) = ${\displaystyle \frac{\frac{\sin x\cos y–\cos x\sin y}{\cos x\cos y}}{\frac{\sin x\sin y+\cos x\cos y}{\cos x\cos y}}}$

= ${\displaystyle \frac{\frac{\sin x\cos y}{\cos x\cos y}–\frac{\cos x\sin y}{\cos x\cos y}}{\frac{\sin x\sin y}{\cos x\cos y}+\frac{\cos x\cos y}{\cos x\cos y}}}$

= ${\displaystyle \frac{\frac{\sin x}{\cos x}–\frac{\sin y}{\cos y}}{\frac{\sin x\sin y}{\cos x\cos y}+1}}$

= ${\displaystyle \frac{\tan x–\tan y}{1+\tan x\tan y}}$

So the formula of tan(x-y) is equal to tan(x-y) = (tanx -tany)/(1+tanx tany).

ALSO READ:

sin(a+b) sin(a-b) Formula

cos(a+b) cos(a-b) Formula

sinx siny Formula

cosx cosy Formula

How to Apply tan(x-y) Formula

Question: Find the value of tan15 degree.

Solution:

To find the value of tan15, one can apply the formula of tan(x-y). Let us put x=45 and y=30 in the formula of tan(x-y) given above. Thus, we have that

tan(45-30) = ${\displaystyle \frac{\tan 45-\tan 30}{1+\tan 45\tan 30}}$

= ${\displaystyle \frac{1-\frac{1}{\sqrt{3}}}{1+1\cdot \frac{1}{\sqrt{3}}}}$

= ${\displaystyle \frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{\frac{\sqrt{3}+1}{\sqrt{3}}}}$

= ${\displaystyle \frac{\sqrt{3}-1}{\sqrt{3}+1}}$

Thus, the value of tan15 degree is equal to (√3-1)/(√3+1) obtained by applying the formula of tan(x-y).

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