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