South Africa
Simplify ^@\sqrt { \dfrac { 1 + cos \theta } { 1 - cos \theta } }^@.
Answer:
^@cosec \theta + cot \theta^@
- 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 ^@.
- 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}^@
