How do you simplify $tan4theta$ to trigonometric functions of a unit $theta$ ?

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Answer 1 · 2016-09-26T10:54:42

$tan4theta=(4tantheta-4tan^3theta)/(1-6tan^2theta+tan^4theta)$

Using the identity $tan2theta=(2tantheta)/(1-tan^2theta)$

$tan4theta=(2tan2theta)/(1-tan^2(2theta)$

= $(2xx(2tantheta)/(1-tan^2theta))/(1-((2tantheta)/(1-tan^2theta))^2$

= $(2xx(2tantheta)/(1-tan^2theta))/(((1-tan^2theta)^2-4tan^2theta)/((1-tan^2theta))^2$

= $(4tantheta)/(1-tan^2theta)xx(1-tan^2theta)^2/((1-tan^2theta)^2-4tan^2theta)$

= $(4tantheta(1-tan^2theta))/((1+tan^4theta-2tan^2theta-4tan^2theta)$

= $(4tantheta-4tan^3theta)/(1-6tan^2theta+tan^4theta)$

How do you simplify tan4thetatan4theta to trigonometric functions of a unit thetatheta?