### If $cot \space \theta + cosec \space \theta = \sqrt { 2 } \space cot \space \theta,$ show that $cot \space \theta - cosec \space \theta = \sqrt { 2 } \space cosec \space \theta$

1. We have $$cot \space \theta + cosec \space \theta = \sqrt { 2 } \space cot \space \theta$$ By squaring both the sides, we get \begin{aligned} &(cot \space \theta + cosec \space \theta)^2 = (\sqrt { 2 } \space cot \space \theta)^2 \\ \implies & cot^2 \space \theta + cosec^2 \space \theta + 2 \space cot \space \theta \space cosec \space \theta = 2 \space cot^2 \space \theta \\ \implies & cosec^2 \space \theta = 2 \space cot^2 \space \theta - cot^2 \space \theta - 2 \space cot \space \theta \space cosec \space \theta \\ \implies & cosec^2 \space \theta + cosec^2 \space \theta = cot^2 \space \theta - 2 \space cot \space \theta \space cosec \space \theta + cosec^2 \space \theta \space \space[\text{ Adding } cosec^2 \space \theta \text{ on both the sides. }] \\ \implies & 2 \space cosec^2 \space \theta = ( cot \space \theta - cosec \space \theta )^2 \\ \implies & cot \space \theta - cosec \space \theta = \sqrt{ 2 } \space cosec \space \theta \\ \end{aligned}
2. Hence, $cot \space \theta - cosec \space \theta = \sqrt{ 2 } \space cosec \space \theta$.