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Prove the identity: (cos x + sin x)^2 + (cos x - sin x)^2 = 2

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lbawa | Student, Undergraduate | eNoter

Posted April 23, 2011 at 9:49 AM via web

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Prove the identity:

(cos x + sin x)^2 + (cos x - sin x)^2 = 2

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted April 23, 2011 at 10:18 AM (Answer #1)

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The identity you had given to be proved was : (cos x + sin x)^2 + (cos x*sin x)^2 = 2

(cos x + sin x)^2 + (cos x*sin x)^2 = 2

opening the brackets gave

=> (cos x)^2 + (sin x)^2 + 2*sin x*cos x + (cos x)^2*(sin x)^2

=> 1 + 2*sin x*cos x + (cos x)^2*(sin x)^2

It was not possible to do anything further to prove that the above was equal to 2.

I think there was a typo, the correct identity should have been : (cos x + sin x)^2 + (cos x - sin x)^2 = 2, the appropriate change has been made.

The correct identity is (cos x + sin x)^2 + (cos x - sin x)^2 = 2

opening the brackets, we get

(cos x)^2 + (sin x)^2 + 2*sin x*cos x + (cos x)^2 + (sin x)^2 - 2*sin x*cos x

use (cos x)^2 + (sin x)^2 = 1

=> 1 + 2*sin x*cos x + 1 - 2*sin x*cos x

cancel the common terms

=> 2

The correct identity which has been proved is : (cos x + sin x)^2 + (cos x - sin x)^2 = 2

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