Question

if cotA+cosecA=m then prove it sinA=2m/(m^2+1)^2

Answer 1

Answer:

$\sin( \alpha ) = \frac{2m}{ {m}^{2} + 1 } .$

Step-by-step explanation:

$\cot( \alpha ) + \csc( \alpha ) = m \: \: \: \: \: ...(i)\\ \\ { \csc( \alpha ) }^{2} - { \cot( \alpha ) }^{2} = 1 \\ \\ ( \csc( \alpha ) + \cot( \alpha )) ( \csc( \alpha ) - \cot( \alpha )) = 1 \\ \\ \csc( \alpha ) - \cot( \alpha ) = \frac{1}{ \cot( \alpha ) + \csc( \alpha ) } = \frac{1}{m} \: \: \: \: \: ...(ii) \\ \\ from \: (i) \: and \: \: (ii) \\ \\ 2 \csc( \alpha ) = m + \frac{1}{m} \\ \\ \frac{2}{ \sin( \alpha ) } = \frac{ {m}^{2} + 1 }{m} \\ \\ \sin( \alpha ) = \frac{2m}{ {m}^{2} + 1 } .$