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December 4, 2016
Posted by **samuel r. edward** on Wednesday, October 12, 2011 at 5:03am.

- trig -
**Damon**, Wednesday, October 12, 2011 at 5:26amYou have a typo which makes it kind of confusing.

tan x = n tan y

so

sin x/cos x = n sin y/cos y

n = (cos y/cos x)(sin x/sin y)

but

m = (sin x/sin y)

sin y = (1/m) sin x

cos^2 y = 1-sin^2y = 1-(1/m^2) sin^2x

so

n = (cos y / cos x)m

n^2 = (m^2)(cos^2 y/cos^2 x)

n^2 = (m^2/cos^2x) [ 1-(1/m^2)sin^2 x]

n^2 = m^2/cos^2x - sin^2x/cos^2x

n^2 cos^2x = m^2 - (1-cos^2x)

n^2 cos^2 x = m^2 - 1 + cos^2 x

(n^2-1)cos^2 x = m^2-1

so

cos^2 x = (m^2-1)/(n^2-1)