Express Cosec a in terms of sec a
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[tex] \csc( \alpha ) = \sqrt{\frac{\sec^{2}( \alpha )}{\sec^{2}( \alpha ) - 1}} [/tex]
Step-by-step explanation:
[tex] \csc( \alpha ) = \frac{1}{ \sin( \alpha ) } [/tex]
[tex] \sec( \alpha ) = \frac{1}{ \cos( \alpha ) } [/tex]
we know that ,
[tex] { \sin^{2}( \alpha ) } + \cos^{2}( \alpha ) = 1[/tex]
so,
[tex] \frac{1}{ \csc^{2}( \alpha )} + \frac{1}{\sec^{2}( \alpha )} = 1[/tex]
[tex] \frac{1}{ \csc^{2}( \alpha )} = 1 - \frac{1}{\sec^{2}( \alpha )} [/tex]
[tex] \frac{1}{ \csc^{2}( \alpha )} = \frac{\sec^{2}( \alpha ) - 1}{\sec^{2}( \alpha )}[/tex]
[tex] { \csc^{2}( \alpha )} = \frac{\sec^{2}( \alpha )}{\sec^{2}( \alpha ) - 1}[/tex]
square root on both sides
[tex] \csc( \alpha ) = \sqrt{\frac{\sec^{2}( \alpha )}{\sec^{2}( \alpha ) - 1}} [/tex]