Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier Malcata

      that may be solved for cos θ as

(2.375)

      if Eq. (2.371) is subtracted from Eq. (2.372), then one gets

      (2.376)

      which gives rise to

(2.377)

      after isolation of sin θ. By the same token, one gets

(2.378)

      after raising both sides of Eq. (2.372) to the nth power, or else

(2.379)

once reciprocals are taken of both sides; Eq. (2.379) may then be rewritten as

      (2.380)

      given the rule of composition of powers – where combination with Eq. (2.369) yields

(2.381)

together with Eq. (2.373) upon replacement of θ by nθ. Since Eq. (2.378) may be rewritten as

(2.382)

      as per Eqs. (2.369) and (2.372), one concludes that

(2.383)

following ordered addition of Eqs. (2.381) and (2.382), together with cancelation of symmetrical terms afterward; by the same token, ordered subtraction of Eq. (2.381) from Eq. (2.382) generates

(2.384)

In view of Eq. (2.375), one may calculate the power of a cosine via

(2.385)

and Eq. (2.377) similarly supports

(2.386)

after retrieving Newton’s binomial as per Eq. (2.236), it is possible to reformulate Eq. (2.385) to

      (2.387)

      where the powers of z and of its reciprocal may be lumped to yield

(2.388)

If the exponent of the cosine function is an even integer, say, 2n, then Eq. (2.388) can be redone to

(2.389)

      after replacement of n by 2n as upper limit, and concomitant replacement of i by 2i as counting variable of the summation – with subsequent splitting of the said summation, so as to make the median term appear explicitly. At this stage, it is convenient to revisit Eq. (2.240) and realize that

      (2.390)

      following straightforward algebraic manipulation; in other words,

(2.391)

      – i.e. the row entries of Pascal’s triangle are symmetrical relative to its median (see Table 2.1). On the other hand, one may introduce a new counting variable satisfying

(2.392)

so that the second summation in Eq. (2.389) can be algebraically converted to

(2.393)

In view of Eq. (2.391), one may reformulate Eq. (2.393) to

      (2.394)