If (1-cos2x)/(1+cos2x) = tan^2 (x). Show that the exact value of tan 22.5 = root2 -1.

Expert Answers

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It is given that : `(1 - cos 2x)/(1+cos2x) = tan^2x`

Let x = 22.5

`tan^2 22.5 = (1 - cos 45)/(1+cos 45)`

= `(1 - 1/sqrt 2)/(1 + 1/sqrt 2)`

= `(sqrt 2 - 1)/(sqrt 2 + 1)`

= `(sqrt 2 - 1)^2/(2 - 1)`

= `(sqrt 2 - 1)^2`

If `tan^2 22.5 = (sqrt 2 - 1)^2` , `tan 22.5 = sqrt 2 - 1`

This proves that `tan 22.5 = sqrt 2 - 1`

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