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Prove that:tan4x=(4tanx(1-tan^2x))/(1-6t...

Prove that:`tan4x=(4tanx(1-tan^2x))/(1-6tan^2x+tan^4x)`

Text Solution

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LHS. `=tan4x=tan2.(2x)`
`=(2tan2x)/(1-(tan2x)^(2)) = ((2.2tanx)/(1-tan^(2)x))/(1-[(2tanx)/(1-tan^(2)x)]^(2)`
`=((4tanx)/(1-tan^(2)x))/((1-tan^(2)x)^(2)-4tan^(2)x)/(1-tan^(2)x)^(2)`
`=(4tanx)/(1-tan^(2)x) xx (1-tan^(2)x)^(2)/(1-2tan^(2)x+tan^(4)x-4tan^(2)x)`
`=4tan x(1-tan^(2)x)/(1-6tan^(2)x+tan^(4)x) =` RHS Hence Proved.
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