Simplify ^@\sqrt { \dfrac { 1 + cos \theta } { 1 - cos \theta } }^@.


Answer:

^@cosec \theta + cot \theta^@

Step by Step Explanation:
  1. We know that ^@1 - cos^2 \theta = sin^2 \theta ^@. Using this identity we can simplify the denominator by multiplying it by ^@1 + cos \theta ^@.
  2. Now, on multiplying numerator and denominator by ^@1 + cos \theta ^@,
    ^@\begin{align} \sqrt { \dfrac { 1 + cos \theta } { 1 - cos \theta } } & = \sqrt { \dfrac { 1 + cos \theta } { 1 - cos \theta } \times \dfrac { 1 + cos \theta } { 1 - cos \theta } } \\ & = \sqrt { \dfrac { (1 + cos \theta)^2 } { 1 - cos^2 \theta } } \\ & = \sqrt { \dfrac { (1 + cos \theta)^2 } { sin ^2 \theta } } \\ & = \dfrac { 1 + cos \theta } { sin \theta } \\ & = \dfrac { 1 } { sin \theta } + \dfrac { cos \theta } { sin \theta } \\ & = cosec \theta + cot \theta \end{align}^@

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