If $\displaystyle\frac{\sec\theta - \tan\theta}{\sec\theta+\tan\theta} = \displaystyle\frac{36}{49}$, find $\displaystyle\frac{cosec\theta - \sec\theta}{cosec\theta + \sec\theta}$

(sec - tan)/(sec + tan) = 36/49. Using (sec^2 - tan^2) = 1, we have (sec - tan)^2 = 36/49, so sec - tan = 6/7. Then sec + tan = 49/6 / 7 = 7/6. Solving these gives sec = (6/7 + 7/6)/2 = 85/84 and tan = (7/6 - 6/7)/2 = 13/84. Using sin/cos and 1/cos, we find cosec = 85/13. The expression (cosec - sec)/(cosec + sec) = (85/13 - 85/84) / (85/13 + 85/84) = (1/13 - 1/84) / (1/13 + 1/84) = (84-13)/(84+13) = 71/97.