Question

√1+cosA/√1-cosA= cosecA + cot A​

Answer 1

Step-by-step explanation:

see in the attachment above

Answer 2

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Q) Prove that :

$\sf \: \dfrac{ \sqrt{1 + cos \: A} }{ \sqrt{1 - cos \: A} } = cosec \: A + cot \: A$

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$\tt \: L. H. S. :\\ \\ \colon \implies \sf \: \sqrt{ \frac{1 + cos \:A}{1 - cos \: A} } \\ \\ \tt \: Rationalise \: the \: denominator: \: \\ \\ \colon \implies \: \sf \sqrt{ \frac{(1 + cos \: A) \times (1 + cos \: A)}{(1 - cos \: A) \times (1 + cos \: A)} } \\ \\ \colon \implies \sf \: \sqrt{ \frac{ {(1 + cos \: A)}^{2} }{ {1}^{2} - {cos}^{2} \: A} } \\ \\ \colon \implies \sf \: \frac{(1 + cos \: A)}{ \sqrt{1 - {cos}^{2} \: A} } \\ \\ \colon \implies \sf \: \frac{1 + cos \: A}{ \sqrt{ {sin}^{2} \: A } } \\ \\ \colon \implies \: \sf \frac{1 + cos \: A}{sin \: A} \\ \\ \colon \implies \sf \: \frac{1}{sin \: A} + \frac{cos \: A}{sin \: A} \\ \\ \colon \implies \underline{\boxed { \pink{ \tt \: cosec \: A + cot \: A}}} \\ \\ { \underline{ \boxed{ \purple { \bf{L.H.S.= R. H. S. }}}}} \: \: \rm \therefore \: Proved$

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$\purple{ \boxed{ \boxed{ \begin{array}{c} \green{\bull \rm \:1 - {cos}^{2} \: \alpha = {sin}^{2} \: \alpha } \\ \\ \red{ \bull \rm \: (x + y)(x - y) = {x}^{2} - {y}^{2} } \\ \\ \pink{ \rm \bull \: \dfrac{1}{sin \: \alpha } = cosec \: \alpha } \\ \\ \blue{ \rm \bull \: \dfrac{cos \: \alpha }{sin \: \alpha } = cot \: \alpha }\end{array}}}}$

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