42. \( \cos \left(\frac{\pi}{2}-x\right)=\sin x \)

This is a well-known trigonometric identity called a “co-function” identity. It states that the cosine of the complement of an angle is equal to the sine of the angle. In other words, for any real number x,   cos(π/2 – x) = sin x. Here's a brief explanation of why this identity holds: 1. On the unit circle, an angle x has the coordinates (cos x, sin x). The complementary angle (π/2 – x) corresponds to “rotating” in such a way that the sine and cosine coordinates swap roles. As a result, cos(π/2 – x) gives the sine of the original angle x. 2. You can also think of it in the context of right triangles. In a right triangle, the acute angles are complementary (they add up to π/2). The cosine of one acute angle is the sine of the other acute angle. Both explanations lead to the conclusion that for every angle x, the identity holds:   cos(π/2 – x) = sin x. Thus, the expression you provided is correct and is true for all values of x.