Question

express all trigonometric ratio in terms of cosec theta

Answer 1

$$\begin{gathered} \sin \mathrm{\theta }=\frac{1}{\mathrm{cosec}} \\ \cos =\sqrt{1-{\sin }^2\mathrm{\theta }}=\sqrt{1-\frac{1}{{\mathrm{cosec}}^2\mathrm{\theta }}}=\sqrt{\frac{{\mathrm{cosec}}^2\mathrm{\theta }-1}{{\mathrm{cosec}}^2\mathrm{\theta }}}=\frac{\sqrt{{\mathrm{cosec}}^2\mathrm{\theta }-1}}{\mathrm{cosec}\mathrm{\theta }} \\ \sec =\frac{1}{\cos \mathrm{\theta }}=\frac{\mathrm{cosec}\mathrm{\theta }}{\sqrt{{\mathrm{cosec}}^2\mathrm{\theta }-1}} \\ \tan =\frac{\sin \mathrm{\theta }}{\cos \mathrm{\theta }}=\frac{{\displaystyle \frac{1}{\mathrm{cosec}\mathrm{\theta }}}}{\frac{\sqrt{{\mathrm{cosec}}^2\mathrm{\theta }-1}}{\mathrm{cosec}\mathrm{\theta }}}=\frac{1}{\sqrt{{\mathrm{cosec}}^2\mathrm{\theta }-1}} \\ \cot =\sqrt{{\mathrm{cosec}}^2\mathrm{\theta }-1} \end{gathered}$$