let I_n = ∫_0 ^(π/4) (dx/(cos^(2n+1) )) (n∈N) 1) find a and b fromR /∀x∈[0,(π/4)] (1/(cosx))=((acosx)/(1−sinx)) +((bcosx)/(1+sinx)) .find I_0 2) verify the relation (1/(cos^(2n+3) x))=(1/(cos^(2n+1) x)) +((sinx sinx)/(cos^(2n+3) )) .find the relation of recurrence between I_n and I


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

$l e t I_{n} = \int_{0}^{\frac{π}{4}} \frac{d x}{{c o s}^{2 n + 1}} (n \in N)$ $1) f i n d a a n d b f r o m R / \forall x \in [0, \frac{π}{4}]$ $\frac{1}{c o s x} = \frac{a c o s x}{1 - s i n x} + \frac{b c o s x}{1 + s i n x} . f i n d I_{0}$ $2) v e r i f y t h e r e l a t i o n$ $\frac{1}{{c o s}^{2 n + 3} x} = \frac{1}{{c o s}^{2 n + 1} x} + \frac{s i n x s i n x}{{c o s}^{2 n + 3}} . f i n d t h e r e l a t i o n$ $o f r e c u r r e n c e b e t w e e n I_{n} a n d I_{n + 1} .$
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