Question

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Answer 1

Answer:

$\sqrt{ \frac{1 + sin \theta}{1 - sin \theta} } + \sqrt{ \frac{1 - sin \theta}{1 + sin \theta} } = 2sec \theta \\ \\ \\ = > \sqrt{ \frac{ ({1 + sin \theta)}^{2} + ( {1 - sin \theta)}^{2} }{(1 - sin \theta)(1 + sin \theta)} } \\ \\ \\ = > \sqrt{ \frac{1 + {sin}^{2} \theta + 2sin \theta + 1 + { \sin }^{2} \theta - 2sin \theta}{1 - {sin}^{2} \theta } } \\ \\ \\ = > \sqrt{ \frac{2 + 2 {sin}^{2} \theta }{cos {}^{2} \theta } } \\ \\ = > \sqrt{ \frac{2 + 2(1 - {cos}^{2} \theta)}{ {cos}^{2} \theta} } \\ \\ = > \sqrt{ \frac{4 - 2 {cos}^{2} \theta }{ {cos}^{2} \theta } } \\ \\ \\ = > 2sec \theta - \sqrt{2}$

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Answer 2

Answer:

$$\begin{lgathered}\sqrt{ \frac{1 + sin \theta}{1 - sin \theta} } + \sqrt{ \frac{1 - sin \theta}{1 + sin \theta} } = 2sec \theta \\ \\ \\ = > \sqrt{ \frac{ ({1 + sin \theta)}^{2} + ( {1 - sin \theta)}^{2} }{(1 - sin \theta)(1 + sin \theta)} } \\ \\ \\ = > \sqrt{ \frac{1 + {sin}^{2} \theta + 2sin \theta + 1 + { \sin }^{2} \theta - 2sin \theta}{1 - {sin}^{2} \theta } } \\ \\ \\ = > \sqrt{ \frac{2 + 2 {sin}^{2} \theta }{cos {}^{2} \theta } } \\ \\ = > \sqrt{ \frac{2 + 2(1 - {cos}^{2} \theta)}{ {cos}^{2} \theta} } \\ \\ = > \sqrt{ \frac{4 - 2 {cos}^{2} \theta }{ {cos}^{2} \theta } } \\ \\ \\ = > 2sec \theta - \sqrt{2}\end{lgathered}$$

Step-by-step explanation:

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