Prove that for all `ninN` `Cosalpha+cos(alpha+beta)+cos(alpha+2beta)+ . . . +cos[alpha+(n-1)beta]`= `(cos[alpha+((n-1)/(2))beta]"sin"((nbeta)/(2)))/("sin"(beta)/(2))`

Explanation

Let P (n): `Cosalpha+cos(alpha+beta)+cos(alpha+2beta)+ . . . +cos[alpha+(n-1)beta]`
`(cos[alpha+((n-1)/(2))beta]"sin"((nbeta)/(2)))/("sin"(beta)/(2))`
Step I We observe that P(1)
`(cos[alpha+((1-1)/(2))]beta"sin"(beta)/(2))/("sin"(beta)/(2))=(cos(alpha+0)"sin"(beta)/(2))/("sin"(beta)/(2))`
`cosalpha=cosalpha`
Hence, P(1) is true.
Step II Now, assume that P(n) is true for n=k.
`P(k):cosalpha+cos(alpha+beta)+cos(alpha+2beta)+. . . +cos[alpha+(k-1)beta]`
`(cos[alpha+((k-1)/(2))]beta"sin"(kbeta)/(2))/("sin"(beta)/(2))`
Step III Now, to prove P(k+1) is true, we have to show that
`P(k):cosalpha+cos(alpha+beta)+cos(alpha+2beta)+. . . +cos[alpha+(k-1)beta]`
`+cos[alpha+1-1beta]=(cos(alpha+(kbeta)/(2))"sin"(k+1)(beta)/(2))/("sin"(beta)/(2))`
LHS `=cosalpha+cos(alpha+beta)+cos(alpha+2beta)+. . . +cos[alpha+(k-1)beta]+cos(alpha+kbeta)`
`=(cos[alpha+((k-1)/(2))beta]"sin"(kbeta)/(2))/("sin"(beta)/(2))+cos(alpha+kbeta)`
`=(cos[alpha+((k-1)/(2))beta]"sin"(kbeta)/(2)+cos(alpha+kbeta)"sin"(beta)/(2))/("sin"(beta)/(2))`
`=("sin"(alpha(kbeta)/(2)-(beta)/(2)+(kbeta)/(2))-"sin"(alpha+(kbeta)/(2)-(beta)/(2)-(kbeta)/(2))+"sin"(alpha+kbeta+(beta)/(2))-"sin"(alpha+kbeta-(beta)/(2)))/(2"sin"(beta)/(2)`)
`=("sin"(alpha+kbeta+(beta)/(2))-"sin"(alpha-(beta)/(2)))/(2"sin"(beta)/(2))`
`=("2cos"(1)/(2)(alpha+(beta)/(2)+kbeta+alpha-(beta)/(2))"sin"(1)/(2)(alpha+(beta)/(2)+kbeta-alpha+(beta)/(2)))/(2"sin"(beta)/(2))`
`=("cos"(((2alpha)+kbeta)/(2))"sin"((kbeta+beta)/(2)))/("sin"(beta)/(2))=("cos"(alpha+(kbeta)/(2))"sin"(k+1)(beta)/(2))/("sin"(beta)/(2))=RHS`
So, P(k+1) is,Hence, P(n)is true.