Consider the integral where n is an integer. Using the trigonometric identity 1 + tan 2 x

Question:

Consider the integral

In 70/4 0 tan"x d.x

where n is an integer. Using the trigonometric identity 1 + tan2x = sec2x, show that

In + In-2 = CTUA So 0 tan" 2x secx dx

and hence obtain the recurrence relation

1= 1 n-1 1. 1-2

Use this to find

7/4 (a)  0 tan x dx (b) TC14 0 tan7x dx

(Recurrence relations of this type are often called reduction formulae, since they provide a systematic way of reducing the value of the parameter n so that a difficult integral may be reduced to an easier one.)

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