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Question Number 28608 by abdo imad last updated on 27/Jan/18

$$transformtanp-tanqthenfindthevalueof$$$$\sum _{k=1}^n\frac{1}{cos\left(k\theta \right)cos((k+1)\theta }.\theta \in R.$$

Answered by Tinkutara last updated on 28/Jan/18

$$\frac{1}{\cos k\theta \cos (k+1)\theta }$$$$=\frac{\cos [(k+1)\theta -k\theta ]}{\cos k\theta \cos (k+1)\theta \cos \theta }$$$$=\frac{\cos (k+1)\theta \cos \theta +\sin (k+1)\theta \sin \theta }{\cos k\theta \cos (k+1)\theta \cos \theta }$$$$=\frac{1+\tan (k+1)\theta \tan \theta }{\cos \theta }$$$$=\frac{\tan (k+1)\theta -\tan \theta }{\sin \theta }$$$$Telescopingitbecomes$$$$\underset{k=1}{\sum ^n}\frac{\tan (k+1)\theta -\tan \theta }{\sin \theta }$$$$=\frac{\tan (n+1)\theta -\tan \theta }{\sin \theta }$$

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