[tex]sin^2(x)[csc^2(x)+sec^2(x)]=sec^2(x) \\\\[-0.35em] ~\dotfill\\\\ sin^2(x)[csc^2(x)+sec^2(x)]\implies sin^2(x)\left[\cfrac{1}{sin^2(x)}+\cfrac{1}{cos^2(x)} \right] \\\\\\ ~~\begin{matrix} sin^2(x) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~ \left[\cfrac{cos^2(x)+sin^2(x)}{~~\begin{matrix} sin^2(x) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~ cos^2(x)} \right]\implies \cfrac{1}{cos^2(x)}\implies sec^2(x)[/tex]
