Question

cosec a(1-cos a) (cosec a+cot a) 1​

Answer 1

To Prove :

cosec A ( 1 - cos A ) ( cosec A + cot A ) = 1

Identities :

${sin}^{2} + {cos}^{2} = 1 \\ i.e. {sin}^{2} = 1 - {cos}^{2} \\ \\ cosec = \frac{1}{sin} \\ \\ cot = \frac{cos}{sin}$

Solution :

$LHS = \cosec(1 - \cos) (cosec + cot) \\ \\ \: \: \: \: = \: \frac{1}{sin} (1 - cos)( \frac{1}{sin} + \frac{cos}{sin} ) \\ \\ \: \: \: \: \: \: = \: \frac{1 - cos}{sin} \times \frac{1 + cos}{sin} \\ \\ numerator \: \: is \: \: in \: \: the \: \: form \: \: of \: \: (x - y)(x + y) = {x}^{2} - {y}^{2} \\ \\ \: \: \: \: = \: \frac{ {1}^{2} - {cos}^{2} }{ {sin}^{2} } \\ \\ \: \: \: \: = \: \frac{ {sin}^{2} }{ {sin}^{2} } \\ \\ \: \: \: \: = \: 1 \\ \\ \: \: \: \: = \: RHS$

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