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Simplify : sin theta{1/sin theta- 1/cosec theta}

User Jefflunt
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To simplify the expression
\displaystyle\sf \sin\theta \left( (1)/(\sin\theta) - (1)/(\csc\theta) \right), we can start by simplifying the denominator terms.

Recall that
\displaystyle\sf \csc\theta = (1)/(\sin\theta). Substituting this into the expression, we have:


\displaystyle\sf \sin\theta \left( (1)/(\sin\theta) - (1)/((1)/(\sin\theta)) \right)

Simplifying further:


\displaystyle\sf \sin\theta \left( (1)/(\sin\theta) - (1)/((1)/(\sin\theta)) \right) = \sin\theta \left( (1)/(\sin\theta) - (\sin\theta)/(1) \right)

Combining the fractions:


\displaystyle\sf \sin\theta \left( (1 - \sin\theta)/(\sin\theta) \right)

Canceling out the common factor of
\displaystyle\sf \sin\theta:


\displaystyle\sf (1 - \sin\theta)/(\sin\theta)

Therefore, the simplified expression is
\displaystyle\sf (1 - \sin\theta)/(\sin\theta).


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User Trajectory
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