Question

if tan^2 theta =1-e^2 then show that sec theta + tan^3theta cosec theta= (2-e^2)^3\2

Answer 1

Dear student

$$\begin{gathered} {\tan }^2\mathrm{\theta }=1-{\mathrm{e}}^2 \\ \mathrm{Consider}, \\ \mathrm{sec\theta }+{\tan }^3\mathrm{\theta cosec\theta } \\ =\sqrt{1+{\tan }^2\mathrm{\theta }}+{\tan }^2\mathrm{\theta }\times \mathrm{tan\theta }\sqrt{1+{\cot }^2\mathrm{\theta }} \\ =\sqrt{2-{\mathrm{e}}^2}+{\left(1-{\mathrm{e}}^2\right)}^{\frac{3}{2}}\sqrt{1+\frac{1}{1-{\mathrm{e}}^2}} \\ =\sqrt{2-{\mathrm{e}}^2}+{\left(1-{\mathrm{e}}^2\right)}^{\frac{3}{2}}\sqrt{\frac{1-{\mathrm{e}}^2+1}{1-{\mathrm{e}}^2}} \\ =\sqrt{2-{\mathrm{e}}^2}+\left(1-{\mathrm{e}}^2\right)\sqrt{2-{\mathrm{e}}^2} \\ =\sqrt{2-{\mathrm{e}}^2}\left(2-{\mathrm{e}}^2\right) \\ ={\left(2-{\mathrm{e}}^2\right)}^{\frac{3}{2}} \end{gathered}$$
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