Show that tanx= (sinx/ cosx)

can be written as:
tan(x-y) = (tanx - tany) / (1+ tanxtany)

write tan (x-y)

= sin(x-y)/cos(x-y)
= [sinxcosy - cosxsiny[/[cosxcosy + sinxsiny]

Now divide everybody by cosxcosy and it will all fall into place.

To show that tan(x) = sin(x) / cos(x) can be written as tan(x - y) = (tan(x) - tan(y)) / (1 + tan(x)tan(y)), we need to utilize the trigonometric identities.

1. Start with the expression: tan(x) = sin(x) / cos(x).

2. Rewrite sin(x) and cos(x) in terms of tan(x) using reciprocal identities:
- sin(x) = (tan(x) * cos(x)) / 1
- cos(x) = 1 / tan(x)

Substituting these values into the initial expression, we get:
tan(x) = [(tan(x) * cos(x)) / 1] / (1 / tan(x)).
Simplifying further gives:
tan(x) = tan(x) * cos(x) / 1.

3. Now, we'll rewrite cos(x) as 1 / tan(x) using the reciprocal identity:
cos(x) = 1 / tan(x).

Substituting this back into the previous expression, we have:
tan(x) = tan(x) * (1 / tan(x)) / 1.
Simplifying further gives:
tan(x) = 1.

Therefore, we can conclude that the given expression tan(x) = sin(x) / cos(x) holds true.

However, it's important to note that the equation tan(x - y) = (tan(x) - tan(y)) / (1 + tan(x)tan(y)) is not equivalent to tan(x) = sin(x) / cos(x).