Question

A derivation of the tangent difference identity is:
tan(x - y) = ?

Answer 1

The tangent difference identity for tan(x - y) is derived using the sum and difference formulas for sine and cosine, resulting in the formula tan(x - y) = (tan(x) - tan(y)) / (1 + tan(x)tan(y)).

Step-by-step explanation:

The question is asking for the tangent difference identity, which expresses tan(x - y) in terms of the tangents of x and y. The tangent difference identity is derived from the sum and difference formulas for sine and cosine, and can be written as:

tan(x - y) = (tan(x) - tan(y)) / (1 + tan(x)tan(y))

To derive this identity, we can use the sine and cosine difference identities:

1. 
   
3. 
   

Since tan(x) = sin(x)/cos(x), we can divide the sine difference identity by the cosine difference identity to obtain the formula for tan(x - y). Through this process, the terms sin(x)/cos(x) and sin(y)/cos(y) are recognized as the tangents of x and y, respectively.

So, by dividing sin(x - y) by cos(x - y), you get:

(sin(x)cos(y) - cos(x)sin(y)) / (cos(x)cos(y) + sin(x)sin(y))

Then, you can split this into:

(sin(x)/cos(x))/(cos(y)/cos(y)) - (cos(x)/cos(x))/(sin(y)/cos(y))

Which simplifies to:

tan(x)/1 - 1/tan(y)

Therefore, the final form after simplifying is the tangent difference identity.