How Can I Derive A General Formula For The Sum Of The Squares Of The Sines And Cosines Of The Angles In An Arithmetic Progression, I.e., ∑[sin^2(α + Kd) + Cos^2(α + Kd)], Where K Ranges From 1 To N, Α Is The Initial Angle, D Is The Common Difference, And N Is The Number Of Terms In The Progression?

To derive a general formula for the sum of the squares of the sines and cosines of the angles in an arithmetic progression, we start with the expression:

${ \sum_{k=1}^{n} \left[ \sin^2(\alpha + kd) + \cos^2(\alpha + kd) \right] }$

We use the trigonometric identity ${\sin^2 \theta + \cos^2 \theta = 1}$ for any angle ${\theta}$. Applying this identity to each term in the sum, we get:

${ \sin^2(\alpha + kd) + \cos^2(\alpha + kd) = 1 }$

Thus, each term in the sum simplifies to 1. Since there are ${n}$ terms, the entire sum is:

${ \sum_{k=1}^{n} 1 = n }$

Therefore, the general formula for the sum is:

${ \boxed{n} }$