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Prove the identity: (tan^2x/1+tan^2x) + (cot^2x/1+cot^2x)= (1-2sin^2xcos^2x)/sinxcosx

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lbawa | Student, Undergraduate | eNoter

Posted April 28, 2011 at 11:27 AM via web

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Prove the identity:

(tan^2x/1+tan^2x) + (cot^2x/1+cot^2x)=

(1-2sin^2xcos^2x)/sinxcosx

Tagged with math, trig identities

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted April 28, 2011 at 12:23 PM (Answer #1)

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The identity that has to be proved is :(tan x)^2 / ( 1 + (tan x)^2) + (cot x)^2 /(1 + (cot x)^2) = (1 - 2*(sin x)^2 (cos x)^2)/sin x cos x

Start with the left hand side:

(tan x)^2 / ( 1 + (tan x)^2) + (cot x)^2 /(1 + (cot x)^2)

=> (tan x)^2 / ( 1 + (tan x)^2) + (1/tan x)^2 /(1 + (1/tan x)^2)

=> (tan x)^2 / ( 1 + (tan x)^2) + [1/(tan x)^2]/[(1 + tan x)^2)/(tan x)^2]

=> (tan x)^2 / ( 1 + (tan x)^2) + 1/(1 + tan x)^2)

=> ((tan x)^2 + 1) / ( 1 + (tan x)^2)

=> 1

The right hand side:

(1 - 2*(sin x)^2 (cos x)^2)/sin x cos x

=>  (1/ sin x cos x) - 2*sin x cos x

=> 2/sin 2x - sin 2x

this cannot be equated to 1.

We do not have an identity here.

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