Find all real numbers $a$ for which there exist different real numbers $b, c, d$ 
different from $a$ such that the four tangents drawn to the curve $y = \sin (x)$ at the points $(a, \sin (a)), 
(b, \sin (b)), (c, \sin (c))$ and $(d, \sin (d))$ form a rectangle.
1. To solve the problem, we start by considering the tangents to the curve \( y = \sin(x) \) at the points \( (a, \sin(a)), (b, \sin(b)), (c, \sin(c)), \) and \( (d, \sin(d)) \). The slope of the tangent to the curve \( y = \sin(x) \) at any point \( (x, \sin(x)) \) is given by the derivative of \( \sin(x) \), which is \( \cos(x) \).

2. For the tangents at these points to form a rectangle, the slopes of the tangents at these points must satisfy the condition of perpendicularity. This means that the product of the slopes of the tangents at any two adjacent points must be \(-1\). Therefore, we have the following conditions:
   \[
   \cos(a) \cos(b) = -1, \quad \cos(b) \cos(c) = -1, \quad \cos(c) \cos(d) = -1, \quad \cos(d) \cos(a) = -1
   \]

3. Let's analyze these conditions. If \( |\cos(a)| < 1 \), then \( |\cos(a) \cos(b)| < 1 \), which contradicts the condition \( \cos(a) \cos(b) = -1 \). Therefore, the only possible values for \( \cos(a) \) are \( 1 \) or \( -1 \).

4. If \( \cos(a) = 1 \), then \( a = 2k\pi \) for some integer \( k \). Similarly, if \( \cos(a) = -1 \), then \( a = (2k+1)\pi \) for some integer \( k \).

5. Since \( a, b, c, \) and \( d \) are different real numbers, and they must all satisfy the same conditions for their cosines, we conclude that \( a, b, c, \) and \( d \) must be multiples of \( \pi \) such that they are distinct.

The final answer is \( \boxed{ a = n\pi } \) for some integer \( n \).