Question

√1-cosA/√1+cosA=cosecA-cotA
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Answer 1

$\sf \underline{Question:-}$

$\tt \: Prove \: \: that \: \: \sqrt{ \frac{1 - cos \:A }{1 + \: cos \: A} } = cosec \: A \: - cot \: \: A$

$\sf \underline {\: Answer:-}$

$LHS \:$

$\implies \bf \: \sqrt{ \frac{1 - cos \: A}{1 + cos \: A} }$

Rationalising the denominator,

$\implies \bf \: \sqrt{ \frac{1 - cos \: A \times 1 - cos \:A }{1 + cos \: A \times 1 - cos \: A} }$

$\implies \bf \: \sqrt{ \frac{ {(1 - cos \:A) }^{2} }{1 - {cos}^{2} A} }$

we know that,

1- cos ² A = sin²A

now,

$\implies \bf \: \sqrt{ \frac{ {(1 - cos \:A) }^{2} }{ {sin}^{2} A} }$

$\implies \bf \: \frac{1 - cos \: A}{sin \: A}$

$\implies \bf \: \frac{1}{sin \: A} - \frac{cos \: A}{sin \: A}$

we know that,

• 1/ sin A = cosec A
• cos A/ sin A = cot A

now,

$\implies \bf \: cosec \: \: A - cot \: \: A$

$\bf \red{LHS = RHS}$