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Question Number 30771 by abdo imad last updated on 25/Feb/18

l e t I_{n} = \int_{0}^{\frac{π}{4}} \frac{d x}{{c o s}^{2 n + 1}} (n \in N)

1) f i n d a a n d b f r o m R / \forall x \in [0, \frac{π}{4}]

\frac{1}{c o s x} = \frac{a c o s x}{1 - s i n x} + \frac{b c o s x}{1 + s i n x} . f i n d I_{0}

2) v e r i f y t h e r e l a t i o n

\frac{1}{{c o s}^{2 n + 3} x} = \frac{1}{{c o s}^{2 n + 1} x} + \frac{s i n x s i n x}{{c o s}^{2 n + 3}} . f i n d t h e r e l a t i o n

o f r e c u r r e n c e b e t w e e n I_{n} a n d I_{n + 1} .