If cotθ = √7, then the value of $$\frac{{{\text{cose}}{{\text{c}}^2}\theta - {{\sec }^2}\theta }}{{{\text{cose}}{{\text{c}}^2}\theta + {{\sec }^2}\theta }}$$   is:

Solution (By Examveda Team)

$$\eqalign{ & \cot \theta = \sqrt 7 \cr & \frac{{{\text{cose}}{{\text{c}}^2}\theta - {{\sec }^2}\theta }}{{{\text{cose}}{{\text{c}}^2}\theta + {{\sec }^2}\theta }} \cr & = \frac{{{{\cos }^2}\theta - {{\sin }^2}\theta }}{{{{\cos }^2}\theta + {{\sin }^2}\theta }} \cr & {\text{Divide by }}{\sin ^2}\theta \cr & = \frac{{\frac{{{{\cos }^2}\theta }}{{{{\sin }^2}\theta }} - \frac{{{{\sin }^2}\theta }}{{{{\sin }^2}\theta }}}}{{\frac{{{{\cos }^2}\theta }}{{{{\sin }^2}\theta }} + \frac{{{{\sin }^2}\theta }}{{{{\sin }^2}\theta }}}} \cr & = \frac{{{{\cot }^2}\theta - 1}}{{{{\cot }^2}\theta + 1}} \cr & = \frac{{7 - 1}}{{7 + 1}} \cr & = \frac{6}{8} \cr & = \frac{3}{4} \cr} $$