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 Math Help > Trigonometry > Trig Equivalences > cos(2x) = (1-tan²(x))/(1+tan²(x))

The "Universal" substitutions in my Summary of Trig Identities, a.k.a. Weierstrass t-substitutions, where t=tan(x/2) are

 formula Weierstrass t-substitution proof tan(x) = 2 tan(x/2)/(1-tan²(x/2)) tan(x) = 2t/(1-t�) from the tan-of-sum formula, using tan(x/2 + x/2) cos(x) = (1-tan�(x/2))/(1+tan�(x/2)) cos(x) = (1-t�)/(1+t�) see below sin(x) = cos(x)tan(x) = 2 tan(x/2)/(1+tan�(x/2)) sin(x) = 2t/(1+t�) by combining the two formulas above.

# cos(x) = (1-tan²(x/2))/(1+tan²(x/2))

There are a number of ways to prove this, some more straightforward than the one I'm about to present.  But this one has a is interesting because it begins with a fascinating perfect square:

1 + 4x²/(1-x²)² =
((1-x²)²+4x²) / (1-x²)² =
(1-2x²+x4+4x²) / (1-x²)² =
(x4+2x²+1) / (1-x²)² =
(x²+1)² / (x²-1)²

We know

tan x+y = (tan x + tan y) / (1 - tan x tan y) (proof and more info)

So it follows that

tan(2x) = 2 tan(x)/(1-tan²(x)), and also
tan(x) = 2 tan(x/2)/(1-tan²(x/2))

We also know that cos²x+sin²x=1, and by dividing this through by cos²x, we get

1+tan²x=1/cos²x, so

cos²x = 1/(1+tan²x)

cos²x = 1/(1+ 4tan²(x/2)/(1-tan²(x/2))² ) -- by substituting tan(x) = 2 tan(x/2)/(1-tan²(x/2))

cos²x = 1/( (1+tan²(x/2))²/(1-tan²(x/2))² ) -- from the "interesting perfect square", above.

cos x = (1-tan²(x/2)) / (1+tan²(x/2))

### Similar formula for sin(x)

We just showed that

cos(x) = (1-tan²(x/2))/(1+tan²(x/2))

And we also pointed out, earlier, that

tan(x) = 2 tan(x/2)/(1-tan²(x/2))

sin(x) = cos(x)tan(x), so to express sin(x) in terms of tan(x/2), we just find the product of the right-hand-sides of the two equations immediately above.  The factor (1-tan²(x/2)) cancels beautifully, so we get

sin(x) = 2 tan(x/2)/(1+tan²(x/2))

(These are variously called the Weierstrass t-substitutions, where t=tan(x/2), and the "Universal" substitutions)

### Related Pages in this website

Trig identities

The Tan of Sum Formula -- tan x+y = (tan x + tan y) / (1 - tan x tan y)

The webmaster and author of this Math Help site is Graeme McRae.