The values for sine and cosine are equal at 45 degrees because they are related to the right triangle formed by the angle.
In a right triangle, the sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. The cosine of an angle is equal to the length of the adjacent side divided by the length of the hypotenuse.
When the angle is 45 degrees, the triangle formed is a 45-45-90 triangle, this is a special triangle in which all the angles are 45 degrees and all the sides are in the ratio of 1:1:√2.
In this triangle, the hypotenuse is √2 times the length of either leg. So, the sine of 45 degrees is opposite/hypotenuse = (1/√2) and the cosine of 45 degrees is adjacent/hypotenuse = (1/√2). Since the opposite and adjacent sides are of the same length in the 45-45-90 triangle, the sine and cosine of 45 degrees are equal.
So, sine 45 = cosine 45 = 1/√2.