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