Question

Verify the identity tan(???? − ????) = tan(????)−tan(????) / 1+tan (????) tan(????) for all (???? − ????) ≠ ???? / 2 + ????n.

Answer 1

Answer:

See the proof below.

Step-by-step explanation:

For this case we need to proof the following identity:

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

So we need to begin with the definition of tan, we know that $tan x = \frac{sin x}{cos x}$ and we have this:

$tan (x-y) = \frac{sin(x-y)}{cos(x-y)}$   (1)

We also have the following identities:

$sin (a-b) = sin a cos b - sin b cos a$

$cos(a-b) = sin a sin b + cos a cos b$

And if we apply those identities into equation (1) we got:

$tan(x-y) = \frac{sin x cos y - sin y cos x}{sin x sin y + cos x cos y}$   (2)

We can divide numerator and denominator from expression (2) by $\frac{1}{cos x cos y}$ like this:

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

And if we simplity we got:

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

And with that we complete the proof. And that appies for all $(x-y) \neq \frac{\pi}{2} +n\pi$