Question

simplify root over((1+sin theta)/(1-sin theta)+(1-sin theta)/1+sin theta))=2sec theta
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Answer 1

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solution~

$\sqrt{\frac{1+sin\theta}{1-sin\theta}}+\sqrt{\frac{1-sin\theta}{1+sin\theta}}=2sec\theta$

$\sqrt{\frac{1+sin\theta}{1-sin\theta}}+\sqrt{\frac{1-sin\theta}{1+sin\theta}}$

➪$\frac{\left(\sqrt{1+sin\theta}\right)^{2}+\left(\sqrt{1-sin\theta}\right)^{2}}{\sqrt{(1-sin\theta)(1+sin\theta)}}$

➪$\frac{1-sin\theta+1+sin\theta}{\sqrt{1^{2}-sin^{2}\theta}}$

➪$\frac{2}{\sqrt{cos^{2}\theta}}$

➪$\frac{2}{cos\theta}$

➪$2sec\theta$

Therefore,

$\sqrt{\frac{1+sin\theta}{1-sin\theta}}+\sqrt{\frac{1-sin\theta}{1+sin\theta}}=2sec\theta$

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