Answer: The required simplified value of the given expression is 2.
Step-by-step explanation: We are given to simplify the following trigonometric expression :
[tex]T_E=(\sin\theta-\cos\theta)^2+(\sin\theta+\cos\theta)^2~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We will be using the following formulas :
[tex](i)~(a-b)^2=a^2-2ab+b^2,\\\\(ii)~(a+b)^2=a^2+2ab+b^2,\\\\(iii)~\sin^2\theta+\cos^2\theta=1.[/tex]
From (i), we have
[tex]T_E\\\\=(\sin\theta-\cos\theta)^2+(\sin\theta+\cos\theta)^2\\\\=\sin^2\theta-2\sin\theta\cos\theta+\cos^2\theta+\sin^2\theta+2\sin\theta\cos\theta+\cos^2\theta\\\\=2(\sin^2\theta+\cos^2\theta)\\\\=2\times1\\\\=2.[/tex]
Thus, the required simplified value of the given expression is 2.
