Question

Show that cos(2x) = (1-t^2)/(1+t^2) when t = tan x

Answer 1

cos(2x) = cos^2x - sin^2x

(1 - tan^2x)/1 + tan^2x 1 + tan^2x = sec^2x
(1-tan^2x)/ sec^2x sec^2x = 1/cos^2x
(1 - tan^2x)/ 1/cos^2x divide by a fraction means * by reciprocal
(1 - tan^2x) * cos^2x distribute
cos^2x - cos^2xtan^2x tan^2x = sin^2x/cos^2x
cos^2x - cos^2x(sin^2x)/cos^2x cos^2x cancel
cos^2x - sin^2x

Answer 2

since the fractions here are rather complex, i may explain it to you by images, ok?

Answer 3

Above is the verification of the equation. Here is another way to prove it.

By drawing the right triangle, if t = tanx, then sinx = t/[sqrt(1+t^2)] and cosx = 1/[sqrt(1+t^2)]. So

cos(2x) = cos^(x) - sin^2(x)
= 1/(1+t^2) - (t^2)/(1+t^2)
= (1-t^2)/(1+t^2)

Answer 4

can you see the picture