Final answer:
In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse, and the sine of an angle is the ratio of the opposite side to the hypotenuse. By using the Pythagorean identity, cos'^2 x + sin'^2 x = 1. Thus, cos' x + sin' x = 1.
Step-by-step explanation:
Let's consider a right triangle. The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse (cos θ = adjacent/hypotenuse), and the sine of an angle is defined as the ratio of the opposite side to the hypotenuse (sin θ = opposite/hypotenuse).
So, cos'^2 x = (cos x)^2, and sin'^2 x = (sin x)^2. Adding these expressions, we have: cos'^2 x + sin'^2 x = (cos x)^2 + (sin x)^2 = 1 (according to the Pythagorean identity).
Therefore, cos'^2 x + sin'^2 x = 1, and when we take the square root of both sides, we get: cos' x + sin' x = √1 = 1. Hence, cos' x + sin' x = 1.