Tangent Fractions Identities

Today's question comes from Yahoo Answers, and it includes trigonometric identities, fractions, and some hard work. Stay with me here, it will take some time.

Prove the identity:
(1 - tan x)/(1 + tan x) = cos 2x / (1 + sin 2x)

As you know, when working out an identity, you need to pick a side and work out until you reach the other side. Since the left side is more complex, I'll work from there to the simple right side. Here we go:

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

Using the tangent quotient (tan x = sin x / cos x), we can write the fraction as:
(1 - sin x / cos x)(1 + sin x / cos x)
(cos x / cos x - sin x / cos x)/(cos x / cos x + sin x / cos x)
((cos x - sin x)/cos x)/((cos x + sin x)/cos x)

Notice that both denominators are cos x, so they can be cancelled:
(cos x - sin x)/(cos x + sin x)

Multiply both parts of the fraction by (cos x + sin x)
(cos x - sin x)(cos x + sin x)/(cos x + sin x)²

Distribute the parentheses:
(cos² x - sin² x)/(cos²2 + 2sinxcosx + sin² x)

As you should know, cos² x - sin² x = cos 2x and cos² x + sin² x = 1, so:
cos 2x / (1 + 2sinxcosx)

As you should also know, 2sinxcosx = sin 2x, so:
cos 2x / (1 + sin 2x)

Q.E.D.

Nadav

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