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

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