Question

If θ lies in the second quadrant, then show that `sqrt((1 - sin theta)/(1 + sin theta)) + sqrt((1 + sin theta)/(1 - sin theta))` = −2sec θ

Answer 1

We have

`sqrt((1 - sin theta)/(1 + sin theta)) + sqrt((1 + sin theta)/(1 - sin theta)) = (1 - sin theta)/sqrt(1 - sin^2theta) + (1 + sin theta)/sqrt(1 - sin^2theta)`

= `2/sqrt(cos^2theta)`

= `2/|cos theta|`  .....(Since `sqrt(alpha^2)` = |α| for every real number α)

Given that θ lies in the second quadrant

So |cos θ| = – cos θ .....(Since cos θ < 0).

Hence, the required value of the expression is `2/(-costheta) = - 2 sectheta`