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


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Answer:

^@-2 cot^2 \theta^@

Step by Step Explanation:
  1. Let,
    ^@S = ^@ ^@\dfrac{ 1 }{ 1 + sec \theta } + \dfrac {1} { 1 - sec \theta }^@
  2. On adding two fractions,
    ^@\begin{align} & \implies S = \dfrac { (1 - sec \theta) + (1 + sec \theta) } { (1 + sec \theta)(1 - sec \theta) } \\ & \implies S = \dfrac {2} { 1 - sec^2 \theta } \end{align}^@
  3. Using identity: ^@sec^2 \theta - 1 = tan^2 \theta,^@
    ^@\begin{align} & \implies S = \dfrac {2} { (-tan^2 \theta) } \\ & \implies S = -2 cot^2 \theta \end{align}^@

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