Question

prove that root 1+costita /1-costita=cosec tita + cot tita

Answer 1

To Prove : $\sf \sqrt{\dfrac{1 + cos \theta}{1 - cos \theta}} = cosec \theta + cot \theta$

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By Multiplying 1 + cosθ in both Numerator and denominator :

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$\twoheadrightarrow\sf \sqrt{\dfrac{1 + cos \theta}{1 - cos \theta} \times \dfrac{1 + cos \theta}{1 + cos \theta}}\\\\\\ \twoheadrightarrow\sf \sqrt{\dfrac{(1 + cos \theta)^2}{1 - cos^2 \theta}}\\\\\\ \twoheadrightarrow\sf \sqrt{\dfrac{(1 + cos \theta)^2}{sin^2 \theta}}\\\\\\ \twoheadrightarrow\sf \dfrac{(1 + cos \theta)}{sin \theta}\\\\\\ \twoheadrightarrow\sf \dfrac{1 }{sin \theta} + \dfrac{cos \theta}{sin \theta}\\\\\\ \twoheadrightarrow{\underline{\boxed{\sf{cosec \theta + cot \theta}}}}$