Question

find the sum of the series 1 + cos theta into cos theta + cos squared theta into cos 2 theta + cos cube theta into cos 3 theta + dot dot dot to alpha

Answer 1

Solution:

The given series is

S= 1 + cos theta into cos theta + cos squared theta into cos 2 theta + cos cube theta into cos 3 theta + dot dot dot to alpha

S=

   $1 + Cos ^2 A + Cos^4 A +Cos^6 A +Cos^8 A+.......$

As this is an infinite geometric series,so sum of infinite geometric series is equal to (A geometric progression having a As first term and r as a common ratio) is $\frac{a}{1-r}$

S=

  $\frac{1}{1-Cos^2A}\\\\ =\frac{1}{Sin^2A}\\\\ = Cosec^2A$

Used the identity, $Sin^2A + Cos^2A=1$