Find all values of $$\theta $$ in the interval $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$ satisfying the equation $$\left( {1 - \tan \,\theta } \right)\left( {1 + \tan \,\theta } \right)\,\,{\sec ^2}\theta + \,\,{2^{{{\tan }^2}\theta }} = 0.$$

Let n and k be positive such that $$n \ge {{k(k + 1)} \over 2}$$ . The number of solutions $$\,({x_1},\,{x_2},\,.....{x_k}),\,{x_1}\,\, \ge \,1,\,{x_2}\, \ge \,2,.......,{x_k} \ge k$$, all integers, satisfying $${x_1} + {x_2} + \,..... + {x_k} = n,\,$$ is......................................

Using mathematical induction prove that for every integer $$n \ge 1,\,\,\left( {{3^{2n}} - 1} \right)$$ is divisible by $${2^{n + 2}}$$ but not by $${2^{n + 3}}$$.

For any odd integer $$n$$ $$ \ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3} = ........$$