A) (a) \[{{\left( 1-{{e}^{2}} \right)}^{\frac{3}{2}}}\]
B) (b) \[{{\left( 2-{{e}^{2}} \right)}^{\frac{1}{2}}}\]
C) (c) \[{{\left( 2-{{e}^{2}} \right)}^{\frac{3}{2}}}\]
D) (d) None of these
Correct Answer: A
Solution :
[a] \[ta{{n}^{2}}\theta =1-{{e}^{2}}\] \[\phi sec\theta +ta{{n}^{3}}\theta .cosec\theta \] \[=sec\theta ta{{n}^{2}}\theta .\text{ }tan\theta .cosec\theta \] \[=sec\theta +ta{{n}^{2}}\theta .\text{ }sec\theta \] \[=sec\theta \left( 1+ta{{n}^{2}}\theta \right)\] \[=sec\theta .se{{c}^{2}}\theta \] \[=se{{c}^{3}}\theta ={{\left( \sqrt{1+ta{{n}^{2}}\theta } \right)}^{3}}={{(1-{{e}^{2}})}^{\frac{3}{2}}}\]Hence, option [a] is correct.
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