To simplify the expression cos^2(π + x) + cos^2(π/2 - x), we can use the trigonometric identity cos(a + b) = cos(a)cos(b) - sin(a)sin(b) and cos(a - b) = cos(a)cos(b) + sin(a)sin(b).
First, let's expand cos^2(π + x) using the identity cos(π + x) = -cos(x):
cos^2(π + x) = (-cos(x))^2 = cos^2(x)
Next, let's expand cos^2(π/2 - x) using the identity cos(π/2 - x) = sin(x):
cos^2(π/2 - x) = sin^2(x)
So, cos^2(π + x) + cos^2(π/2 - x) simplifies to cos^2(x) + sin^2(x), which is equal to 1 (due to the Pythagorean identity for cos^2(x) + sin^2(x) = 1).
To simplify the expression cos^2(π + x) + cos^2(π/2 - x), we can use the trigonometric identity cos(a + b) = cos(a)cos(b) - sin(a)sin(b) and cos(a - b) = cos(a)cos(b) + sin(a)sin(b).
First, let's expand cos^2(π + x) using the identity cos(π + x) = -cos(x):
cos^2(π + x) = (-cos(x))^2 = cos^2(x)
Next, let's expand cos^2(π/2 - x) using the identity cos(π/2 - x) = sin(x):
cos^2(π/2 - x) = sin^2(x)
So, cos^2(π + x) + cos^2(π/2 - x) simplifies to cos^2(x) + sin^2(x), which is equal to 1 (due to the Pythagorean identity for cos^2(x) + sin^2(x) = 1).
Therefore, the simplified expression is 1.