If $I_m = \int_{\pi/4}^{\pi/2} cot^m \theta d\theta$ then value of I4 + I2 is

Q: If $\large I_m = \int_{\pi/4}^{\pi/2} cot^m \theta d\theta$ then value of I4 + I2 is

(A) 1/2

(B) 1/3

(C) 1/4

(D) none of these

Sol: $\large I_m = \int_{\pi/4}^{\pi/2} cot^m \theta d\theta$

$\large I_m = \int_{\pi/4}^{\pi/2} cot^{m-2} ( cosec^2\theta – 1 ) d\theta$

$\large I_m = \int_{\pi/4}^{\pi/2} cot^{m-2} cosec^2\theta d\theta – I_{m-2}$

$\large I_m = [-\frac{cot^{m-1}\theta}{m-1}]_{\pi/4}^{\pi/2}-I_{m-2}$

$\large I_m + I_{m-2} = \frac{1}{m-1}$

$\large I_4 + I_2 = \frac{1}{3}$

Hence (B) is the correct answer.