a trigonometric identity...?

Question

Subtract and Simplify:
(sinx-1 / cosx) - (cosx / sinx-1)

Answer 1

2 tan x. Here's how:

First, write the ORIGINAL EXPRESSION as you should have done:

[(sin x - 1) / cosx] - [cosx / ( sin x - 1)].

(Check carefully where the parenthesis signs etc. should really go, to avoid ambiguity.)

Express it as something over [cos x (sin x - 1)] as a common factor. Then it's:

[(sin x - 1)^2 - cos^2 x] / [cos x (sin x - 1)] =

[sin^2 x - 2 sin x +1 - cos^2 x] / [cos x (sin x - 1)];

but 1 - cos^2 x is sin^2 x. Substituting that, the numerator becomes:

2 sin^2 x - 2 sin x = 2 sin x (sin x - 1).

When you now divide by the common factor in the denominator, you're left with 2 sin x / cos x, i.e.:

2 tan x QED.

Live long and prosper.